answer:<think>Alright, so I've got this problem about the Mandelbrot set and fractals, and I need to help an actress with her lighting design. Let me try to break this down step by step.First, the problem has two parts. The first part is about determining the range of values for ( a ) and ( b ) such that the sequence ( {z_n} ) remains bounded after 100 iterations. The second part is about calculating the total length of a spotlight's path based on the Hausdorff dimension of the Mandelbrot set's boundary.Starting with the first part. I remember that the Mandelbrot set is defined as the set of complex numbers ( c = a + bi ) for which the function ( f_c(z) = z^2 + c ) does not escape to infinity when iterated from ( z_0 = 0 ). So, we need to find all ( c ) such that the sequence ( z_{n+1} = z_n^2 + c ) remains bounded.I also recall that the Mandelbrot set is contained within the disk of radius 2 in the complex plane. That is, if the magnitude of ( z_n ) ever exceeds 2, the sequence will definitely escape to infinity. So, for each ( c = a + bi ), we can iterate the function up to 100 times and check if the magnitude of ( z_n ) stays below 2.But the question is asking for the range of ( a ) and ( b ). I think the Mandelbrot set is mostly located in the area where ( a ) is between -2 and 1, and ( b ) is between -1 and 1. But I'm not entirely sure about the exact boundaries. Maybe I should recall the specific regions.The main cardioid of the Mandelbrot set is given by ( c = frac{1 - sqrt{1 - 4(1 - sin^2 theta)}}{2} e^{itheta} ) for ( theta ) in [0, 2π). But that might be too complicated. Alternatively, I remember that the leftmost point of the Mandelbrot set is at ( c = -2 ), and the rightmost point is at ( c = frac{1}{4} ). The top and bottom are symmetric around the real axis, so they should be at ( b = pm 1 ).But wait, is that accurate? Let me think. The Mandelbrot set is symmetric about the real axis, so yes, the imaginary parts will mirror each other. But the exact boundaries are a bit more complex because the set isn't a perfect shape; it has intricate details. However, for the purposes of this problem, maybe we can approximate the range.So, if I were to give a rough range, ( a ) would be between -2 and 1, and ( b ) would be between -1 and 1. But to be more precise, I think the actual range for ( a ) is from -2 to 0.25, because the rightmost point is at ( c = 0.25 ). Wait, no, actually, the rightmost point is at ( c = 0.25 ), but the set extends a bit beyond that in some areas. Hmm, I might be mixing things up.Wait, no, the main cardioid is centered at ( c = 0 ) with a radius of 0.5, but the entire set extends to the left up to ( c = -2 ). So, the real part ( a ) ranges from -2 to approximately 0.25, and the imaginary part ( b ) ranges from -1 to 1. But actually, the exact maximum and minimum for ( b ) might be a bit more than 1 because of the protrusions.But since the problem is about the range such that the sequence remains bounded after 100 iterations, and considering that 100 iterations is a large number, we can approximate that the range is ( a in [-2, 0.25] ) and ( b in [-1, 1] ). However, I think the exact bounds are ( a in [-2, 0.25] ) and ( b in [-1, 1] ), but I should verify.Wait, actually, the Mandelbrot set is bounded within the circle of radius 2, so any ( c ) outside of that circle will definitely escape. So, the maximum modulus of ( c ) is 2, which would mean ( a^2 + b^2 leq 4 ). But that's a rough bound. The actual set is more constrained.But for the purposes of this problem, maybe the answer expects the approximate range of ( a ) and ( b ). So, I think it's safe to say that ( a ) ranges from -2 to 1, and ( b ) ranges from -1 to 1. However, I'm not entirely certain. Maybe I should look up the exact bounds.Wait, no, I can't look things up, but I remember that the Mandelbrot set is approximately contained within the rectangle ( a in [-2, 1] ) and ( b in [-1, 1] ). So, I think that's a reasonable approximation for this problem.Moving on to the second part. The spotlight's path is a fractal with Hausdorff dimension ( D approx 2 ). The length of the path is given by ( L(epsilon) approx K epsilon^{1 - D} ), where ( epsilon = 0.01 ).So, substituting ( D = 2 ) and ( epsilon = 0.01 ), we get:( L(0.01) approx K (0.01)^{1 - 2} = K (0.01)^{-1} = K times 100 ).But wait, the problem says the Hausdorff dimension ( D approx 2 ). However, the Hausdorff dimension of the boundary of the Mandelbrot set is actually 2, which is the same as the dimension of the plane, but it's a fractal curve. Wait, no, the boundary of the Mandelbrot set is a fractal with Hausdorff dimension 2, which is the same as the plane, but it's a curve. Hmm, I might be confusing things.Wait, no, the Hausdorff dimension of the boundary of the Mandelbrot set is actually 2, which is the same as the plane, but it's a fractal curve. So, if the spotlight's path has scaling properties similar to the boundary, then ( D = 2 ).But in the formula, ( L(epsilon) approx K epsilon^{1 - D} ). So, substituting ( D = 2 ), we get ( L(epsilon) approx K epsilon^{-1} ). So, as ( epsilon ) decreases, the length increases as ( 1/epsilon ).But the problem is asking for the total length when ( epsilon = 0.01 ). However, without knowing the constant ( K ), we can't compute the exact numerical value. But maybe the problem expects an expression in terms of ( K ).Wait, the problem says "calculate the total length of the spotlight's path if ( epsilon = 0.01 ) and ( D approx 2 )." So, substituting, we get ( L = K times (0.01)^{1 - 2} = K times (0.01)^{-1} = K times 100 ).But without knowing ( K ), we can't give a numerical answer. Maybe I'm missing something. Perhaps ( K ) is related to the perimeter or something else. Alternatively, maybe the problem expects an expression in terms of ( K ), so the total length is ( 100K ).Alternatively, perhaps the problem is expecting to recognize that with ( D = 2 ), the length diverges as ( epsilon ) approaches zero, but since ( epsilon = 0.01 ), it's finite. But without ( K ), we can't compute it numerically.Wait, maybe I'm misunderstanding the formula. The formula ( L(epsilon) approx K epsilon^{1 - D} ) is for the length of a fractal curve. For a fractal with Hausdorff dimension ( D ), the length scales as ( epsilon^{1 - D} ). So, when ( D = 2 ), the length scales as ( epsilon^{-1} ), which means as ( epsilon ) decreases, the length increases without bound. But in reality, the spotlight's path can't have infinite length, so perhaps ( K ) is a scaling factor that depends on the actual size of the set.But without more information, I think the answer is simply ( L = 100K ). So, the total length is 100 times the constant ( K ).Wait, but maybe I should think differently. If the Hausdorff dimension is 2, which is the same as the plane, then the length might not be defined in the usual sense. But the problem states that the spotlight's path forms a fractal with scaling properties similar to the boundary, so we can use the given formula.So, putting it all together, for the first part, the range of ( a ) is approximately from -2 to 1, and ( b ) from -1 to 1. For the second part, the total length is ( 100K ).But let me double-check the first part. The Mandelbrot set is defined as all ( c ) such that the orbit of 0 under ( z^2 + c ) doesn't escape to infinity. The set is bounded within the disk of radius 2, so ( |c| leq 2 ). But the exact range for ( a ) and ( b ) is more specific.The real part ( a ) ranges from -2 to approximately 0.25, because the rightmost point of the Mandelbrot set is at ( c = 0.25 ). The imaginary part ( b ) ranges from -1 to 1 because of the symmetry and the shape of the set. So, more accurately, ( a in [-2, 0.25] ) and ( b in [-1, 1] ).But wait, is the imaginary part exactly from -1 to 1? I think the top and bottom of the main cardioid are at ( c = pm i ), which are ( a = 0 ), ( b = pm 1 ). So, yes, the imaginary part ranges from -1 to 1.So, to sum up, for the first part, ( a ) is between -2 and 0.25, and ( b ) is between -1 and 1. For the second part, the length is ( 100K ).I think that's the best I can do without more specific information.</think>

answer:<think>Alright, so I have this problem where I need to model the political influence of a leader over a region using a partial differential equation. The equation given is:[frac{partial P}{partial t} = D frac{partial^2 P}{partial x^2} - alpha P + beta cos(omega t)]with boundary conditions ( P(0, t) = 0 ) and ( P(L, t) = 0 ), and the initial condition ( P(x, 0) = P_0(x) ). I need to solve for ( P(x, t) ) and then analyze the maximum influence at a specific time ( T ).First, I remember that this is a linear PDE with constant coefficients, and it's a nonhomogeneous equation because of the ( beta cos(omega t) ) term. The boundary conditions are homogeneous Dirichlet conditions, which suggests that separation of variables might be a good approach here.Let me recall the method of separation of variables. We assume a solution of the form ( P(x, t) = X(x)T(t) ). Plugging this into the PDE, we get:[X(x) frac{dT}{dt} = D T(t) frac{d^2X}{dx^2} - alpha X(x) T(t) + beta cos(omega t)]Hmm, but wait, the nonhomogeneous term ( beta cos(omega t) ) complicates things because it doesn't separate into a function of ( x ) and ( t ). So maybe I need to use a method for nonhomogeneous PDEs, like eigenfunction expansion or Duhamel's principle.Alternatively, perhaps I can rewrite the equation in terms of a steady-state solution and a transient solution. Let me think. If I look for a particular solution ( P_p(x, t) ) that satisfies the nonhomogeneous equation, and a homogeneous solution ( P_h(x, t) ) that satisfies the homogeneous version of the PDE, then the general solution would be ( P(x, t) = P_p(x, t) + P_h(x, t) ).So, first, let me solve the homogeneous equation:[frac{partial P_h}{partial t} = D frac{partial^2 P_h}{partial x^2} - alpha P_h]with boundary conditions ( P_h(0, t) = 0 ) and ( P_h(L, t) = 0 ), and initial condition ( P_h(x, 0) = P_0(x) - P_p(x, 0) ). Wait, but I don't know ( P_p(x, t) ) yet.Alternatively, maybe I should find the particular solution first. Since the nonhomogeneous term is ( beta cos(omega t) ), which is a function of time only, perhaps I can assume a particular solution of the form ( P_p(x, t) = Q(x) cos(omega t) + R(x) sin(omega t) ). Let me try that.Assume ( P_p(x, t) = Q(x) cos(omega t) + R(x) sin(omega t) ). Then, compute the derivatives:[frac{partial P_p}{partial t} = -Q(x) omega sin(omega t) + R(x) omega cos(omega t)][frac{partial^2 P_p}{partial x^2} = Q''(x) cos(omega t) + R''(x) sin(omega t)]Plugging into the PDE:[- Q omega sin(omega t) + R omega cos(omega t) = D (Q'' cos(omega t) + R'' sin(omega t)) - alpha (Q cos(omega t) + R sin(omega t)) + beta cos(omega t)]Now, group the terms by ( cos(omega t) ) and ( sin(omega t) ):For ( cos(omega t) ):[R omega = D Q'' - alpha Q + beta]For ( sin(omega t) ):[- Q omega = D R'' - alpha R]So, we have a system of ODEs:1. ( D Q'' - alpha Q + beta - R omega = 0 )2. ( D R'' - alpha R + Q omega = 0 )This seems a bit complicated. Maybe I can write this as a single equation by differentiating one and substituting into the other.Alternatively, perhaps I can assume that ( R = k Q ) for some constant ( k ), but I'm not sure. Maybe another approach is better.Wait, another idea: since the nonhomogeneous term is only in the ( cos(omega t) ) term, perhaps the particular solution can be written as ( P_p(x, t) = Q(x) cos(omega t) ). Let me try that.So, ( P_p(x, t) = Q(x) cos(omega t) ). Then,[frac{partial P_p}{partial t} = -Q(x) omega sin(omega t)][frac{partial^2 P_p}{partial x^2} = Q''(x) cos(omega t)]Plugging into the PDE:[- Q omega sin(omega t) = D Q'' cos(omega t) - alpha Q cos(omega t) + beta cos(omega t)]Hmm, this gives:- For ( cos(omega t) ): ( D Q'' - alpha Q + beta ) must equal zero because there's no ( cos(omega t) ) on the left side. But wait, the left side has a ( sin(omega t) ) term, which doesn't appear on the right. So, unless ( Q ) is such that the coefficients of ( sin(omega t) ) and ( cos(omega t) ) separately balance.Wait, actually, the equation is:Left side: ( - Q omega sin(omega t) )Right side: ( D Q'' cos(omega t) - alpha Q cos(omega t) + beta cos(omega t) )So, equating coefficients:For ( sin(omega t) ): ( - Q omega = 0 ) which implies ( Q = 0 ), but then the other terms would have to satisfy ( 0 = D Q'' - alpha Q + beta ), which would be ( 0 = 0 + 0 + beta ), which is impossible unless ( beta = 0 ). But ( beta ) is given as a non-zero amplitude. So this approach doesn't work.Therefore, my initial assumption that ( P_p ) is only a cosine term is insufficient. I need to include both sine and cosine terms, as I did earlier.So, going back to the system:1. ( D Q'' - alpha Q + beta - R omega = 0 )2. ( D R'' - alpha R + Q omega = 0 )Let me try to solve this system. Let me write them as:1. ( D Q'' - alpha Q - R omega = -beta )2. ( D R'' - alpha R + Q omega = 0 )This is a coupled system of ODEs. Maybe I can express one variable in terms of the other.From equation 2: ( D R'' - alpha R = - Q omega )From equation 1: ( D Q'' - alpha Q = R omega - beta )So, let me substitute ( R ) from equation 2 into equation 1.Wait, equation 2 can be written as ( R'' = (alpha R - Q omega)/D )Similarly, equation 1: ( Q'' = (alpha Q + R omega - beta)/D )This seems complicated, but maybe I can write this as a single fourth-order ODE.Alternatively, perhaps assume that ( Q ) and ( R ) are proportional, like ( R = k Q ), but I don't know if that's valid.Alternatively, perhaps take the derivative of equation 2 and substitute.Wait, let me try to express ( R ) from equation 2.From equation 2: ( D R'' = alpha R - Q omega )So, ( R'' = (alpha/D) R - ( omega / D ) Q )Similarly, from equation 1: ( D Q'' = alpha Q + R omega - beta )So, ( Q'' = (alpha/D) Q + ( omega / D ) R - ( beta / D ) )Now, let me take the derivative of equation 2:( R''' = (alpha/D) R' - ( omega / D ) Q' )But I don't see an immediate way to relate this to equation 1.Alternatively, perhaps substitute ( R ) from equation 2 into equation 1.Wait, equation 2 can be rearranged to express ( Q ) in terms of ( R ):From equation 2: ( Q = (D R'' + alpha R)/ omega )Plug this into equation 1:( D Q'' - alpha Q - R omega = -beta )Substitute ( Q = (D R'' + alpha R)/ omega ):First, compute ( Q'' ):( Q = (D R'' + alpha R)/ omega )So, ( Q' = (D R''' + alpha R') / omega )( Q'' = (D R'''' + alpha R'') / omega )Now, plug into equation 1:( D (D R'''' + alpha R'') / omega - alpha (D R'' + alpha R)/ omega - R omega = -beta )Multiply through by ( omega ) to eliminate denominators:( D (D R'''' + alpha R'') - alpha (D R'' + alpha R) - R omega^2 = -beta omega )Simplify term by term:1. ( D^2 R'''' + D alpha R'' )2. ( - D alpha R'' - alpha^2 R )3. ( - R omega^2 )Combine like terms:- ( D^2 R'''' )- ( D alpha R'' - D alpha R'' = 0 )- ( - alpha^2 R - R omega^2 )So, the equation becomes:( D^2 R'''' - (alpha^2 + omega^2) R = - beta omega )This is a fourth-order ODE for ( R(x) ). Hmm, this seems quite involved. Maybe there's a simpler approach.Wait, perhaps instead of assuming a particular solution in terms of sine and cosine, I can use the method of eigenfunction expansion. Since the homogeneous equation is similar to the heat equation with decay, perhaps the eigenfunctions are sine functions due to the boundary conditions.Let me recall that for the equation ( frac{partial P}{partial t} = D frac{partial^2 P}{partial x^2} - alpha P ), the eigenfunctions would satisfy ( -D phi'' = lambda phi ) with ( phi(0) = phi(L) = 0 ). So, the eigenfunctions are ( phi_n(x) = sin(n pi x / L) ) with eigenvalues ( lambda_n = (D (n pi / L)^2 - alpha) ).Wait, actually, the eigenvalue equation is ( -D phi'' = lambda phi ), so ( phi'' = - (lambda / D) phi ). The solutions are sines and cosines, but with boundary conditions ( phi(0) = phi(L) = 0 ), so only sine terms survive. Thus, ( phi_n(x) = sin(n pi x / L) ) and ( lambda_n = D (n pi / L)^2 ).But in our case, the PDE is ( frac{partial P}{partial t} = D frac{partial^2 P}{partial x^2} - alpha P + beta cos(omega t) ). So, it's a nonhomogeneous equation. Therefore, the solution can be expressed as a Fourier series in terms of the eigenfunctions ( phi_n(x) ).Let me write the solution as:[P(x, t) = sum_{n=1}^{infty} c_n(t) sinleft( frac{n pi x}{L} right)]Then, plug this into the PDE:[sum_{n=1}^{infty} frac{dc_n}{dt} sinleft( frac{n pi x}{L} right) = D sum_{n=1}^{infty} c_n left( -frac{n^2 pi^2}{L^2} right) sinleft( frac{n pi x}{L} right) - alpha sum_{n=1}^{infty} c_n sinleft( frac{n pi x}{L} right) + beta cos(omega t)]Now, the nonhomogeneous term ( beta cos(omega t) ) is not in terms of the eigenfunctions, so I need to express it as a Fourier series as well. However, since it's a cosine function, it might not have a sine expansion unless we extend it as an odd function, but that might complicate things.Alternatively, perhaps I can project both sides onto each eigenfunction ( sin(m pi x / L) ) to find the coefficients ( c_n(t) ).So, multiply both sides by ( sin(m pi x / L) ) and integrate from 0 to L:[int_0^L sum_{n=1}^{infty} frac{dc_n}{dt} sinleft( frac{n pi x}{L} right) sinleft( frac{m pi x}{L} right) dx = int_0^L left[ -D sum_{n=1}^{infty} c_n frac{n^2 pi^2}{L^2} sinleft( frac{n pi x}{L} right) - alpha sum_{n=1}^{infty} c_n sinleft( frac{n pi x}{L} right) + beta cos(omega t) right] sinleft( frac{m pi x}{L} right) dx]Using orthogonality of sine functions, the left side becomes:[frac{dc_m}{dt} cdot frac{L}{2}]The right side becomes:1. For the first term: ( -D frac{n^2 pi^2}{L^2} c_n ) integrated against ( sin(n pi x / L) sin(m pi x / L) ), which gives ( -D frac{n^2 pi^2}{L^2} c_n cdot frac{L}{2} delta_{nm} )2. Similarly, the second term gives ( -alpha c_n cdot frac{L}{2} delta_{nm} )3. The third term is ( beta cos(omega t) ) integrated against ( sin(m pi x / L) ). Since ( cos(omega t) ) is a function of time only, the integral becomes ( beta cos(omega t) cdot int_0^L sin(m pi x / L) dx ). But ( int_0^L sin(m pi x / L) dx = frac{L}{m pi} (1 - cos(m pi)) ). Since ( m ) is an integer, ( cos(m pi) = (-1)^m ), so this becomes ( frac{L}{m pi} (1 - (-1)^m) ). For odd ( m ), this is ( frac{2L}{m pi} ), and for even ( m ), it's zero.Therefore, putting it all together:Left side: ( frac{dc_m}{dt} cdot frac{L}{2} )Right side:1. ( -D frac{m^2 pi^2}{L^2} c_m cdot frac{L}{2} )2. ( -alpha c_m cdot frac{L}{2} )3. ( beta cos(omega t) cdot frac{L}{m pi} (1 - (-1)^m) )Simplify:[frac{dc_m}{dt} cdot frac{L}{2} = - frac{D m^2 pi^2}{2 L} c_m - frac{alpha L}{2} c_m + beta cos(omega t) cdot frac{L}{m pi} (1 - (-1)^m)]Divide both sides by ( frac{L}{2} ):[frac{dc_m}{dt} = - frac{D m^2 pi^2}{L^2} c_m - alpha c_m + frac{2 beta}{m pi} (1 - (-1)^m) cos(omega t)]This is an ordinary differential equation for each ( c_m(t) ). The equation is linear and nonhomogeneous. The homogeneous part is:[frac{dc_m}{dt} + left( frac{D m^2 pi^2}{L^2} + alpha right) c_m = 0]The solution to the homogeneous equation is:[c_m^{(h)}(t) = c_m^{(0)} e^{ - left( frac{D m^2 pi^2}{L^2} + alpha right) t }]Now, for the particular solution ( c_m^{(p)}(t) ), since the nonhomogeneous term is ( frac{2 beta}{m pi} (1 - (-1)^m) cos(omega t) ), we can assume a particular solution of the form:[c_m^{(p)}(t) = A_m cos(omega t) + B_m sin(omega t)]Compute the derivative:[frac{dc_m^{(p)}}{dt} = -A_m omega sin(omega t) + B_m omega cos(omega t)]Plug into the ODE:[- A_m omega sin(omega t) + B_m omega cos(omega t) + left( frac{D m^2 pi^2}{L^2} + alpha right) (A_m cos(omega t) + B_m sin(omega t)) = frac{2 beta}{m pi} (1 - (-1)^m) cos(omega t)]Group the terms:For ( cos(omega t) ):[B_m omega + left( frac{D m^2 pi^2}{L^2} + alpha right) A_m = frac{2 beta}{m pi} (1 - (-1)^m)]For ( sin(omega t) ):[- A_m omega + left( frac{D m^2 pi^2}{L^2} + alpha right) B_m = 0]This gives a system of equations:1. ( left( frac{D m^2 pi^2}{L^2} + alpha right) A_m + B_m omega = frac{2 beta}{m pi} (1 - (-1)^m) )2. ( - A_m omega + left( frac{D m^2 pi^2}{L^2} + alpha right) B_m = 0 )We can solve this system for ( A_m ) and ( B_m ). Let me write it in matrix form:[begin{cases}left( frac{D m^2 pi^2}{L^2} + alpha right) A_m + omega B_m = frac{2 beta}{m pi} (1 - (-1)^m) - omega A_m + left( frac{D m^2 pi^2}{L^2} + alpha right) B_m = 0end{cases}]Let me denote ( gamma_m = frac{D m^2 pi^2}{L^2} + alpha ) for simplicity.Then the system becomes:1. ( gamma_m A_m + omega B_m = frac{2 beta}{m pi} (1 - (-1)^m) )2. ( - omega A_m + gamma_m B_m = 0 )From equation 2: ( gamma_m B_m = omega A_m ) => ( B_m = frac{omega}{gamma_m} A_m )Plug into equation 1:( gamma_m A_m + omega cdot frac{omega}{gamma_m} A_m = frac{2 beta}{m pi} (1 - (-1)^m) )Simplify:( gamma_m A_m + frac{omega^2}{gamma_m} A_m = frac{2 beta}{m pi} (1 - (-1)^m) )Factor out ( A_m ):( A_m left( gamma_m + frac{omega^2}{gamma_m} right) = frac{2 beta}{m pi} (1 - (-1)^m) )Simplify the term in the parenthesis:( gamma_m + frac{omega^2}{gamma_m} = frac{gamma_m^2 + omega^2}{gamma_m} )Thus,( A_m = frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{gamma_m}{gamma_m^2 + omega^2} )And since ( B_m = frac{omega}{gamma_m} A_m ), we have:( B_m = frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{omega}{gamma_m^2 + omega^2} )Therefore, the particular solution is:[c_m^{(p)}(t) = frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{gamma_m}{gamma_m^2 + omega^2} cos(omega t) + frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{omega}{gamma_m^2 + omega^2} sin(omega t)]This can be written as:[c_m^{(p)}(t) = frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{gamma_m cos(omega t) + omega sin(omega t)}{gamma_m^2 + omega^2}]Notice that ( gamma_m cos(omega t) + omega sin(omega t) ) can be expressed as ( sqrt{gamma_m^2 + omega^2} cos(omega t - delta_m) ), where ( delta_m = arctan(omega / gamma_m) ). But for now, I'll keep it as is.Therefore, the general solution for ( c_m(t) ) is:[c_m(t) = c_m^{(h)}(t) + c_m^{(p)}(t) = c_m^{(0)} e^{ - gamma_m t } + frac{2 beta}{m pi} (1 - (-1)^m) cdot frac{gamma_m cos(omega t) + omega sin(omega t)}{gamma_m^2 + omega^2}]Now, the initial condition is ( P(x, 0) = P_0(x) ). So, at ( t = 0 ), we have:[P(x, 0) = sum_{n=1}^{infty} c_n(0) sinleft( frac{n pi x}{L} right) = P_0(x)]Therefore, the coefficients ( c_n(0) ) are the Fourier sine coefficients of ( P_0(x) ):[c_n(0) = frac{2}{L} int_0^L P_0(x) sinleft( frac{n pi x}{L} right) dx]So, putting it all together, the solution ( P(x, t) ) is:[P(x, t) = sum_{n=1}^{infty} left[ c_n^{(0)} e^{ - gamma_n t } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega t) + omega sin(omega t)}{gamma_n^2 + omega^2} right] sinleft( frac{n pi x}{L} right)]Where ( gamma_n = frac{D n^2 pi^2}{L^2} + alpha ).Now, for part 2, the author wants to analyze the maximum political influence at a specific time ( t = T ). The maximum influence ( P_{text{max}}(T) ) is the maximum value of ( P(x, T) ) over ( x ) in ( [0, L] ).Given the solution above, ( P(x, T) ) is a sum of sine terms with coefficients that depend on ( T ). To find the maximum, we can consider the expression for ( P(x, T) ) and find its maximum value.However, since the solution is a Fourier series, the maximum value will depend on the coefficients and the sine terms. The maximum of a sum of sine functions isn't straightforward, but perhaps we can express ( P_{text{max}}(T) ) as an integral involving the Fourier coefficients.Wait, the problem says to express ( P_{text{max}}(T) ) as an integral involving the Fourier series coefficients of the initial condition ( P_0(x) ). So, perhaps we can write ( P_{text{max}}(T) ) in terms of the initial coefficients ( c_n^{(0)} ) and the particular solution coefficients.But actually, since ( P(x, T) ) is given by the sum above, the maximum value would be the supremum of this function over ( x ). However, expressing this as an integral might not be straightforward. Alternatively, perhaps we can consider the maximum in terms of the amplitude of each Fourier mode.Wait, another approach: the maximum of ( P(x, T) ) occurs where the derivative with respect to ( x ) is zero. But since ( P(x, T) ) is a sum of sine functions, its derivative is a sum of cosine functions. Setting the derivative to zero would give the critical points, but solving this analytically might be difficult.Alternatively, perhaps we can consider the maximum influence as the maximum of the sum of the amplitudes of each term. That is, the maximum value would be the sum of the absolute values of each coefficient times the maximum of the sine function, which is 1. But this would be an upper bound, not necessarily the actual maximum.Wait, but the problem says to express ( P_{text{max}}(T) ) as an integral involving the Fourier series coefficients. Maybe it's referring to the fact that the maximum can be found by integrating the square of the function or something similar, but I'm not sure.Alternatively, perhaps we can consider that the maximum influence is achieved when all the sine terms are in phase, i.e., when their arguments align to maximize the sum. But this would depend on the specific coefficients and the phase shifts, which complicates things.Wait, perhaps a better approach is to note that the maximum of ( P(x, T) ) can be found by considering the function as a sum of sinusoids and using the fact that the maximum of a sum of sinusoids is the square root of the sum of squares of the amplitudes, but this is only true for orthogonal functions, which sine functions are not when considering different frequencies.Wait, actually, for a function expressed as a sum of sinusoids with different frequencies, the maximum is not simply the sum of the amplitudes, but it's more complex. However, if we consider the function at a specific time ( T ), then ( P(x, T) ) is a function of ( x ) only, and its maximum can be found by considering the integral of its square or using some other integral expression.Alternatively, perhaps the problem is referring to expressing the maximum influence in terms of the Fourier coefficients, which are integrals of ( P_0(x) ) against the sine functions. So, ( P_{text{max}}(T) ) can be expressed as an integral involving these coefficients.But I'm not entirely sure. Let me think again.Given that ( P(x, T) ) is a sum of terms involving ( c_n(T) sin(n pi x / L) ), the maximum value of ( P(x, T) ) would be the maximum of this sum over ( x ). However, expressing this maximum as an integral might not be straightforward unless we use some integral representation of the maximum.Alternatively, perhaps the problem is asking to express ( P_{text{max}}(T) ) in terms of the Fourier coefficients, which are integrals of ( P_0(x) ). So, maybe we can write ( P_{text{max}}(T) ) as an integral involving ( c_n^{(0)} ) and the particular solution coefficients.But I'm not sure. Maybe I need to consider that the maximum influence is the sum of the contributions from each mode, each of which has an amplitude that depends on ( c_n^{(0)} ) and the particular solution.Wait, perhaps the maximum influence is achieved when each sine term is at its maximum, which is 1, so the maximum would be the sum of the absolute values of the coefficients. But this is an upper bound, not necessarily the actual maximum.Alternatively, perhaps the maximum influence is given by the integral of ( P(x, T) ) squared, but that would give the energy, not the maximum.Wait, maybe the problem is expecting me to express ( P_{text{max}}(T) ) as the integral of ( P(x, T) ) times some delta function or something, but that seems too abstract.Alternatively, perhaps the maximum influence can be found by considering the Fourier coefficients and the particular solution, and then using some integral expression to find the maximum.Wait, maybe I'm overcomplicating this. The problem says to express ( P_{text{max}}(T) ) as an integral involving the Fourier series coefficients of the initial condition. So, perhaps I can write ( P_{text{max}}(T) ) as the sum over ( n ) of ( |c_n(T)| ), since each term contributes up to ( |c_n(T)| ) to the maximum.But that would be an upper bound, not necessarily the actual maximum. However, if the phases are aligned, the maximum could be the sum of the amplitudes. But since the particular solution introduces phase shifts, it's not clear.Alternatively, perhaps the maximum influence is given by the integral of ( P(x, T) ) over ( x ), but that's not necessarily the maximum.Wait, another idea: the maximum of ( P(x, T) ) can be found by considering the function's expression and then using calculus to find its maximum. However, since it's a sum of sine functions, this might not lead to a simple integral expression.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is related to the initial condition's Fourier coefficients and the particular solution's coefficients, and thus express ( P_{text{max}}(T) ) as an integral involving these coefficients.But I'm not entirely sure. Maybe I should proceed step by step.Given that ( P(x, T) ) is:[P(x, T) = sum_{n=1}^{infty} left[ c_n^{(0)} e^{ - gamma_n T } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} right] sinleft( frac{n pi x}{L} right)]The maximum value of this function over ( x ) would be the maximum of the sum of these sine terms. However, the maximum of a sum of sine functions is not simply the sum of their amplitudes unless they are all in phase.But perhaps, for the purpose of this problem, we can express ( P_{text{max}}(T) ) as the integral over ( x ) of ( P(x, T) ) times the Dirac delta function at the point where the maximum occurs. However, this seems too abstract and not helpful.Alternatively, perhaps the problem is expecting me to consider that the maximum influence is the sum of the absolute values of the coefficients, which would be:[P_{text{max}}(T) = sum_{n=1}^{infty} left| c_n^{(0)} e^{ - gamma_n T } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} right|]But this is an upper bound, not necessarily the actual maximum.Alternatively, perhaps the maximum influence can be expressed as an integral involving the Fourier coefficients by considering the function's maximum in terms of its Fourier series.Wait, another approach: the maximum of ( P(x, T) ) can be found by considering the function's expression and then using the fact that the maximum of a function is equal to the integral of the function multiplied by the Dirac delta function at the maximum point. But this is more of a distributional approach and might not be what the problem is asking for.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is related to the initial condition's energy or something similar, but I'm not sure.Wait, perhaps I should consider that the maximum influence is achieved when the particular solution is at its maximum, and the homogeneous solution has decayed. So, if the homogeneous solution decays over time, the maximum influence at time ( T ) would be dominated by the particular solution.In that case, the maximum influence would be the maximum of the particular solution, which is:[P_p(x, T) = sum_{n=1}^{infty} frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} sinleft( frac{n pi x}{L} right)]The maximum of this function would depend on the coefficients and the sine terms. However, expressing this maximum as an integral involving the Fourier coefficients of the initial condition might not be straightforward.Wait, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the integral of ( P(x, T) ) times some function that picks out the maximum, but I'm not sure.Alternatively, perhaps the problem is referring to the fact that the maximum influence can be expressed as the integral of the initial condition's Fourier coefficients multiplied by some kernel involving time ( T ).Wait, considering that ( P(x, T) ) is expressed as a sum over ( n ) of terms involving ( c_n^{(0)} e^{-gamma_n T} ) and the particular solution terms, perhaps the maximum influence can be expressed as an integral involving ( c_n^{(0)} ) and the particular solution coefficients.But I'm not sure how to proceed. Maybe I should look for a different approach.Wait, perhaps the maximum influence ( P_{text{max}}(T) ) can be found by considering the function ( P(x, T) ) and finding its maximum value. Since ( P(x, T) ) is a sum of sine functions, the maximum occurs where the derivative with respect to ( x ) is zero. However, solving for ( x ) where the derivative is zero would involve setting a sum of cosine terms to zero, which is complicated.Alternatively, perhaps the maximum can be expressed in terms of the Fourier coefficients by considering the function's maximum as the integral of ( P(x, T) ) times the Dirac delta function at the maximum point. But this is more of a formal expression.Wait, another idea: perhaps the maximum influence can be expressed as the integral of ( P(x, T) ) squared, but that would give the energy, not the maximum.Alternatively, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the sum of the absolute values of the Fourier coefficients, which would be an upper bound on the maximum.But I'm not sure. Maybe I should consider that the maximum influence is achieved when all the sine terms are in phase, so the maximum would be the sum of the absolute values of the coefficients. However, this is only an upper bound and not necessarily the actual maximum.Wait, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as an integral involving the Fourier coefficients, which are integrals of ( P_0(x) ) against the sine functions. So, maybe:[P_{text{max}}(T) = int_0^L P(x, T) delta(x - x_{text{max}}) dx]But this is too abstract and not helpful.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is related to the initial condition's Fourier coefficients and the particular solution's coefficients, and thus express ( P_{text{max}}(T) ) as an integral involving these coefficients.But I'm not sure. Maybe I should proceed to the next step.Given that I have the solution ( P(x, t) ), the maximum influence at time ( T ) is the maximum of ( P(x, T) ) over ( x ). To express this as an integral involving the Fourier series coefficients, perhaps I can write:[P_{text{max}}(T) = max_{x in [0, L]} sum_{n=1}^{infty} left[ c_n^{(0)} e^{ - gamma_n T } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} right] sinleft( frac{n pi x}{L} right)]But this is just restating the solution. To express this as an integral, perhaps I can use the fact that the maximum of a function can be expressed using the Dirac delta function:[P_{text{max}}(T) = int_0^L P(x, T) deltaleft( frac{dP}{dx}(x, T) right) dx]But this is a formal expression and might not be what the problem is asking for.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is related to the initial condition's Fourier coefficients and the particular solution's coefficients, and thus express ( P_{text{max}}(T) ) as an integral involving these coefficients.But I'm not sure. Maybe I should consider that the maximum influence is the sum of the absolute values of the Fourier coefficients, which would be:[P_{text{max}}(T) = sum_{n=1}^{infty} left| c_n^{(0)} e^{ - gamma_n T } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} right|]But this is an upper bound, not necessarily the actual maximum.Alternatively, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the integral of ( P(x, T) ) times some function that picks out the maximum, but I'm not sure.Wait, perhaps the problem is referring to the fact that the maximum influence can be expressed as an integral involving the Fourier coefficients by considering the function's maximum in terms of its Fourier series. For example, using the fact that the maximum of a function is related to its Fourier coefficients through some integral expression.But I'm not aware of a standard integral expression for the maximum of a function in terms of its Fourier coefficients. Therefore, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the sum of the absolute values of the Fourier coefficients, which is an upper bound.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is achieved when the particular solution is at its maximum, and the homogeneous solution has decayed. So, if ( T ) is large enough, the homogeneous solution decays to zero, and the maximum influence is determined by the particular solution.In that case, the maximum influence would be the maximum of the particular solution:[P_p(x, T) = sum_{n=1}^{infty} frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} sinleft( frac{n pi x}{L} right)]The maximum of this function would depend on the coefficients and the sine terms. However, expressing this maximum as an integral involving the Fourier coefficients of the initial condition might not be straightforward.Wait, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the integral of ( P(x, T) ) times the Dirac delta function at the point where the maximum occurs. But this is more of a formal expression and not helpful.Alternatively, perhaps the problem is expecting me to recognize that the maximum influence is related to the initial condition's energy or something similar, but I'm not sure.Given that I'm stuck, perhaps I should summarize what I have so far.The solution to the PDE is:[P(x, t) = sum_{n=1}^{infty} left[ c_n^{(0)} e^{ - gamma_n t } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega t) + omega sin(omega t)}{gamma_n^2 + omega^2} right] sinleft( frac{n pi x}{L} right)]Where ( gamma_n = frac{D n^2 pi^2}{L^2} + alpha ), and ( c_n^{(0)} = frac{2}{L} int_0^L P_0(x) sinleft( frac{n pi x}{L} right) dx ).For the maximum influence at time ( T ), ( P_{text{max}}(T) ), it's the maximum of ( P(x, T) ) over ( x ). Expressing this as an integral involving the Fourier coefficients might involve integrating over ( x ) with some weighting function, but I'm not sure.Alternatively, perhaps the problem is expecting me to consider that the maximum influence is achieved when the particular solution is at its peak, and the homogeneous solution has decayed. Therefore, the maximum influence would be determined by the particular solution's coefficients.In that case, the maximum influence would be the maximum of:[P_p(x, T) = sum_{n=1}^{infty} frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} sinleft( frac{n pi x}{L} right)]The maximum of this function would depend on the coefficients and the sine terms. However, to express this maximum as an integral involving the Fourier coefficients of the initial condition, perhaps we can write:[P_{text{max}}(T) = int_0^L P(x, T) deltaleft( frac{dP}{dx}(x, T) right) dx]But this is a formal expression and might not be what the problem is asking for.Alternatively, perhaps the problem is expecting me to express ( P_{text{max}}(T) ) as the sum of the absolute values of the Fourier coefficients, which would be:[P_{text{max}}(T) = sum_{n=1}^{infty} left| c_n^{(0)} e^{ - gamma_n T } + frac{2 beta}{n pi} (1 - (-1)^n) cdot frac{gamma_n cos(omega T) + omega sin(omega T)}{gamma_n^2 + omega^2} right|]But this is an upper bound, not necessarily the actual maximum.Given that I'm not sure, perhaps I should conclude that the maximum influence ( P_{text{max}}(T) ) is given by the sum of the absolute values of the Fourier coefficients, which are integrals of the initial condition ( P_0(x) ) against the sine functions.Therefore, the maximum influence can be expressed as an integral involving the Fourier series coefficients of the initial condition, and it is maximized when the particular solution's coefficients are maximized, which occurs when the denominator ( gamma_n^2 + omega^2 ) is minimized, i.e., when ( gamma_n ) is as small as possible, which happens for smaller ( n ).So, the conditions under which ( P_{text{max}}(T) ) is maximized would involve the particular solution's coefficients being as large as possible, which occurs when ( gamma_n ) is small, i.e., when ( D ) is small or ( alpha ) is small, or when ( n ) is small.But I'm not entirely confident about this conclusion. However, given the time I've spent, I think this is the best I can do.</think>

answer:<think>Okay, so I have this problem about the probability distribution of goals scored in a football match. Let me try to figure it out step by step.First, part 1 asks me to derive the probability distribution function for the total number of goals scored by both teams combined in a 90-minute match. Both teams have constant probabilities of scoring per minute, p_A for team A and p_B for team B. These are independent.Hmm, so each minute, team A can score with probability p_A, and team B can score with probability p_B. Since the match is 90 minutes long, I guess I can model the number of goals each team scores as independent Poisson processes. Because goals are rare events, right? So, the number of goals scored by each team in the match can be modeled using Poisson distributions.Wait, so for team A, the number of goals X_A in 90 minutes would have a Poisson distribution with parameter λ_A = p_A * 90. Similarly, for team B, X_B would have a Poisson distribution with λ_B = p_B * 90. Since the goals are independent, the total number of goals X = X_A + X_B would also be Poisson distributed with parameter λ = λ_A + λ_B.So, the probability distribution function for X is Poisson with λ = 90(p_A + p_B). Therefore, the probability that the total number of goals is k is:P(X = k) = (e^{-λ} * λ^k) / k!And the expected total number of goals is just λ, which is 90(p_A + p_B). That seems straightforward.Now, moving on to part 2. It says that in the Iranian league, matches often end with 3 goals. Given p_A = 0.01 and p_B = 0.01 per minute, compute the exact probability that a match will end with exactly 3 goals.Wait, so using the same model as part 1, but with specific p_A and p_B. So, let's compute λ for this case.First, λ_A = 90 * 0.01 = 0.9, and λ_B = 90 * 0.01 = 0.9. So, total λ = 0.9 + 0.9 = 1.8.Therefore, the total number of goals X is Poisson(1.8). So, the probability that X = 3 is:P(X = 3) = (e^{-1.8} * (1.8)^3) / 3!Let me compute that.First, e^{-1.8} is approximately... Let me calculate that. e^{-1.8} ≈ 0.1653.Then, (1.8)^3 = 1.8 * 1.8 * 1.8. 1.8 * 1.8 is 3.24, and 3.24 * 1.8 is 5.832.So, numerator is 0.1653 * 5.832 ≈ 0.1653 * 5.832. Let me compute that:0.1653 * 5 = 0.82650.1653 * 0.832 ≈ 0.1653 * 0.8 = 0.13224; 0.1653 * 0.032 ≈ 0.00529. So total ≈ 0.13224 + 0.00529 ≈ 0.13753.So total numerator ≈ 0.8265 + 0.13753 ≈ 0.96403.Denominator is 3! = 6.So, P(X = 3) ≈ 0.96403 / 6 ≈ 0.1607.So approximately 16.07% chance.Wait, but is that exact? The question says "compute the exact probability." Hmm, so maybe I should keep it in terms of e^{-1.8} and fractions.Let me write it as:P(X = 3) = (e^{-1.8} * (1.8)^3) / 6But maybe we can express 1.8 as 9/5, so 1.8 = 9/5.Therefore, (1.8)^3 = (9/5)^3 = 729 / 125.So, P(X = 3) = (e^{-1.8} * 729 / 125) / 6 = (e^{-1.8} * 729) / (125 * 6) = (e^{-1.8} * 729) / 750.Simplify 729 / 750: both divisible by 3. 729 ÷ 3 = 243; 750 ÷ 3 = 250. So, 243/250.So, P(X = 3) = (e^{-1.8} * 243) / 250.Is that exact? Well, e^{-1.8} is an irrational number, so we can't express it exactly without approximation, but in terms of e, that's as exact as it gets.Alternatively, if we want to write it as a decimal, we can compute e^{-1.8} more accurately.Let me compute e^{-1.8} more precisely.We know that e^{-1} ≈ 0.3678794412e^{-0.8} ≈ 0.4493288869So, e^{-1.8} = e^{-1} * e^{-0.8} ≈ 0.3678794412 * 0.4493288869Let me compute that:0.3678794412 * 0.4 = 0.14715177650.3678794412 * 0.0493288869 ≈ ?First, compute 0.3678794412 * 0.04 = 0.01471517765Then, 0.3678794412 * 0.0093288869 ≈ approximately 0.003435.So total ≈ 0.01471517765 + 0.003435 ≈ 0.01815So total e^{-1.8} ≈ 0.1471517765 + 0.01815 ≈ 0.1653017765So, e^{-1.8} ≈ 0.1653017765Then, (1.8)^3 = 5.832So, numerator is 0.1653017765 * 5.832 ≈Let me compute that:0.1653017765 * 5 = 0.82650888250.1653017765 * 0.832 ≈Compute 0.1653017765 * 0.8 = 0.13224142120.1653017765 * 0.032 ≈ 0.005290So total ≈ 0.1322414212 + 0.005290 ≈ 0.1375314212So total numerator ≈ 0.8265088825 + 0.1375314212 ≈ 0.9640403037Divide by 6: 0.9640403037 / 6 ≈ 0.16067338395So approximately 0.16067, or 16.067%.So, the exact probability is (e^{-1.8} * (1.8)^3) / 6, which is approximately 16.07%.But since the question asks for the exact probability, maybe we can leave it in terms of e^{-1.8} multiplied by the constants, but I think they might expect the decimal approximation.Alternatively, perhaps they want the expression in terms of factorials or something else.Wait, but in the Poisson formula, it's e^{-λ} * λ^k / k! So, in this case, e^{-1.8} * (1.8)^3 / 6.Alternatively, if we write 1.8 as 9/5, then:e^{-9/5} * (9/5)^3 / 6 = e^{-9/5} * 729 / (125 * 6) = e^{-9/5} * 729 / 750 = e^{-9/5} * 243 / 250.So, that's another exact form.But unless they want it in terms of fractions, maybe the decimal is acceptable.Alternatively, perhaps they want it as a fraction multiplied by e^{-1.8}, but I think the approximate decimal is acceptable for the exact probability.So, summarizing:1. The total number of goals follows a Poisson distribution with λ = 90(p_A + p_B). The expected total is λ.2. For the Iranian league, with p_A = p_B = 0.01 per minute, λ = 1.8. The probability of exactly 3 goals is approximately 16.07%.I think that's it.Final Answer1. The expected total number of goals is boxed{90(p_A + p_B)}.2. The exact probability is boxed{dfrac{e^{-1.8} cdot (1.8)^3}{6}} or approximately boxed{0.1607}.</think>

answer:<think>Okay, so I have this problem where Alex and Jordan want to be sustainable in their new home. They're installing a rainwater harvesting system and solar panels. I need to figure out two things: first, the minimum efficiency their rainwater system needs to have, and second, the minimum area of solar panels required and the total cost to make sure it all fits within 10,000. Let me take this step by step.Starting with the first part about the rainwater harvesting system. The goal is to meet at least 50% of their annual water needs through rainwater. Their daily consumption is 200 gallons, so I should calculate their annual water usage first. Annual water consumption would be 200 gallons/day multiplied by 365 days. Let me write that down: 200 * 365. Hmm, 200 times 300 is 60,000, and 200 times 65 is 13,000, so total is 73,000 gallons per year. They want to meet 50% of that, which is half of 73,000. So, 73,000 divided by 2 is 36,500 gallons. They need to collect at least 36,500 gallons annually through rainwater.Now, the rainwater system can collect up to 0.8 gallons per square foot of roof area per inch of rainfall. Their roof is 1,500 square feet, and the average annual rainfall is 40 inches. So, the total rainwater they can collect is 0.8 * 1500 * 40. Let me compute that: 0.8 times 1500 is 1200, and 1200 times 40 is 48,000 gallons. So, without considering efficiency, they can collect 48,000 gallons.But they need 36,500 gallons. So, the efficiency is the ratio of the required water to the total collected. Efficiency is (required / total) * 100%. So, that's (36,500 / 48,000) * 100. Let me calculate that: 36,500 divided by 48,000. Hmm, 36,500 divided by 48,000 is approximately 0.7604, so multiplying by 100 gives about 76.04%. So, they need an efficiency of at least 76.04%. Since efficiency can't be more than 100%, and they need to meet the requirement, so the minimum efficiency is 76.04%.Wait, let me double-check that. The system can collect 48,000 gallons, but they need 36,500. So, the efficiency is the amount they need divided by what the system can collect. So yes, 36,500 / 48,000 is indeed approximately 76.04%. So, they need their system to be at least 76.04% efficient. I think that's correct.Moving on to the second part about the solar panels. They cost 3 per watt, are 20% efficient, and receive an average of 5 kWh/m²/day. They want to offset 70% of their monthly electricity consumption, which is 900 kWh. So, first, let's figure out how much they need to generate.70% of 900 kWh is 0.7 * 900 = 630 kWh per month. To find the daily requirement, since there are about 30 days in a month, 630 / 30 = 21 kWh per day. So, they need 21 kWh per day from the solar panels.Now, solar panels are 20% efficient and receive 5 kWh/m²/day. So, the energy generated per square meter per day is 5 kWh/m² * 20% efficiency. Wait, no, that's not quite right. The solar irradiance is 5 kWh/m²/day, which is the amount of sunlight hitting the panels. The efficiency is 20%, so the energy produced is 5 * 0.20 = 1 kWh/m²/day.So, each square meter of panels produces 1 kWh per day. They need 21 kWh per day, so the area required is 21 / 1 = 21 m². Therefore, they need at least 21 square meters of solar panels.Now, calculating the cost. The panels cost 3 per watt. Wait, I need to figure out the total power required. Since they need 21 kWh per day, and assuming they use it all in a day, the power required is 21 kWh / 24 hours = 0.875 kW. But wait, that's not the right approach. The panels generate 1 kWh per day per square meter, so 21 kWh per day requires 21 kW? No, that's not correct.Wait, I think I confused power and energy. Let me clarify. The solar panels produce energy, not power. So, if they need 21 kWh per day, and each square meter produces 1 kWh per day, then 21 m² is correct. But to find the power rating, we need to consider the peak power. Wait, maybe I don't need to. The cost is given per watt, so I need to find the total wattage.Wait, perhaps I need to think in terms of the system's capacity. The panels are 20% efficient, but the cost is 3 per watt. So, if I know the area, I can find the power. The power output of solar panels is typically given in watts, which is the maximum power they can produce under standard conditions.But in this case, the efficiency is given as 20%, and the solar irradiance is 5 kWh/m²/day. So, the energy produced per day is 1 kWh/m², as I calculated earlier. So, to get 21 kWh/day, they need 21 m².But to find the power, we need to know the peak sun hours. Wait, maybe I'm overcomplicating. The question says the panels are 20% efficient, and the home receives 5 kWh/m²/day. So, each square meter produces 1 kWh per day.But to find the power, we can use the formula: Energy (kWh) = Power (kW) * Sun Hours. If we assume that the peak sun hours are such that 5 kWh/m²/day is the total, then perhaps the power is related to that.Wait, maybe I should think differently. The total energy needed per day is 21 kWh. Each square meter provides 1 kWh per day, so 21 m² is needed. The power of the panels is the area multiplied by the power per square meter. But the power per square meter depends on the efficiency and the solar irradiance.Wait, maybe I need to calculate the power in terms of peak watts. Solar panels' power is given in watts, which is the maximum power they can produce under standard test conditions (1000 W/m² irradiance, 25°C temperature). But in this case, the average irradiance is 5 kWh/m²/day, which is about 5,000 Wh/m²/day, or approximately 208.33 Wh/m²/hour on average (since 5,000 / 24 ≈ 208.33). But the efficiency is 20%, so the power output per square meter would be 208.33 Wh/m²/hour * 20% = 41.666 Wh/m²/hour, which is 0.041666 kW/m²/hour. To get 21 kWh per day, we need:Energy = Power * Time21 kWh = Power (kW) * 24 hoursPower = 21 / 24 = 0.875 kWBut this power is the total power needed. Since each square meter provides 0.041666 kW, the number of square meters required is 0.875 / 0.041666 ≈ 21 m², which matches our earlier calculation.So, the area needed is 21 m². Now, the cost is 3 per watt. But we need to find the total power in watts. The total power is 0.875 kW, which is 875 watts. So, the cost is 875 * 3 = 2,625.Wait, but hold on. Is the power 875 watts? Because 21 m² at 0.041666 kW/m² is 0.875 kW, which is 875 watts. So, yes, the total power is 875 watts. Therefore, the cost is 875 * 3 = 2,625.But let me make sure. The panels are 20% efficient, and the irradiance is 5 kWh/m²/day. So, each square meter produces 1 kWh per day. They need 21 kWh per day, so 21 m². The power is the area multiplied by the power per square meter. The power per square meter is the efficiency times the irradiance in kW/m². Wait, the irradiance is 5 kWh/m²/day, which is 5,000 Wh/m²/day. Divided by 24 hours, that's about 208.33 Wh/m²/hour. So, power per square meter is 208.33 Wh/m²/hour * 20% = 41.666 Wh/m²/hour, which is 0.041666 kW/m². So, total power is 21 m² * 0.041666 kW/m² ≈ 0.875 kW or 875 watts. So, yes, the cost is 875 * 3 = 2,625.Now, adding the cost of the rainwater system. Wait, the problem mentions that the total cost should fit within 10,000. But it doesn't specify the cost of the rainwater system. Hmm, maybe I need to calculate that as well.Wait, in the first part, we calculated the efficiency required, but we didn't calculate the cost. The problem doesn't give the cost of the rainwater system, so perhaps we need to assume that the cost of the rainwater system is separate and we need to make sure that the solar panels cost plus the rainwater system cost is within 10,000.But the problem doesn't specify the cost of the rainwater system, so maybe we only need to calculate the cost of the solar panels and ensure it's within 10,000. Alternatively, perhaps the rainwater system has a cost based on the efficiency, but the problem doesn't specify that either.Wait, let me re-read the problem. It says: "calculate the minimum efficiency... to achieve this goal." Then for the solar panels, "determine the minimum area... Additionally, calculate the total cost to install these panels, ensuring it fits within their 10,000 budget along with the rainwater system cost."So, the total cost includes both systems. But we don't know the cost of the rainwater system. Hmm, maybe I need to assume that the rainwater system cost is based on the efficiency? Or perhaps it's a fixed cost. The problem doesn't specify, so maybe I need to proceed with the information given.Wait, maybe the rainwater system cost isn't given, so perhaps we can only calculate the solar panel cost and ensure that it's within 10,000. But the problem says "along with the rainwater system cost," so we need to know the total cost.Wait, perhaps the rainwater system cost is based on the efficiency. But the problem doesn't specify any cost per efficiency or per component. Hmm, this is confusing.Wait, maybe I need to proceed with the information given. The solar panels cost 3 per watt, and we calculated that they need 875 watts, costing 2,625. If the total budget is 10,000, then the rainwater system must cost 10,000 - 2,625 = 7,375. But since the problem doesn't specify the cost of the rainwater system, perhaps we can only calculate the solar panel cost and ensure it's within the budget, but the problem says "along with the rainwater system cost," so maybe we need to assume that the rainwater system is free or that we don't have its cost.Wait, perhaps I'm overcomplicating. Maybe the problem just wants the cost of the solar panels and to make sure it's within 10,000, regardless of the rainwater system. But the wording says "along with the rainwater system cost," so perhaps we need to consider both. But without knowing the rainwater system's cost, we can't calculate the total. Maybe the rainwater system is free, or maybe it's a fixed cost. Hmm.Wait, perhaps the rainwater system's cost isn't required for the second part, only the solar panels. Let me check the problem again."Additionally, calculate the total cost to install these panels, ensuring it fits within their 10,000 budget along with the rainwater system cost."So, the total cost is solar panels plus rainwater system, and it should be ≤ 10,000. But we don't know the rainwater system's cost. Hmm, maybe the rainwater system's cost is based on the efficiency? Or perhaps it's a fixed cost, but the problem doesn't specify.Wait, maybe the rainwater system's cost isn't dependent on efficiency, but rather on the components, which aren't given. So, perhaps the problem expects us to only calculate the solar panel cost and ensure it's within the budget, assuming the rainwater system is within the remaining budget. But without knowing the rainwater system's cost, we can't be certain.Wait, maybe the problem expects us to calculate the solar panel cost and then note that it's within the budget, regardless of the rainwater system. But the problem says "along with the rainwater system cost," so perhaps we need to consider both. But since we don't have the rainwater system's cost, maybe we can only calculate the solar panel cost and state that it's 2,625, leaving 7,375 for the rainwater system.But the problem doesn't ask for that. It just asks to calculate the total cost of the solar panels, ensuring it fits within the budget along with the rainwater system. So, perhaps the answer is that the solar panels cost 2,625, and as long as the rainwater system is ≤ 7,375, the total is within 10,000. But the problem doesn't specify the rainwater system's cost, so maybe we can only provide the solar panel cost.Wait, perhaps I'm overcomplicating. Let me proceed with what I have. The solar panels cost 2,625, so as long as the rainwater system is ≤ 7,375, the total is within 10,000. But since the problem doesn't specify the rainwater system's cost, maybe we can only answer the solar panel part.Alternatively, maybe the rainwater system's cost is based on the efficiency. But the problem doesn't specify any cost per efficiency or per component. Hmm.Wait, perhaps the rainwater system's cost is fixed, and we don't need to calculate it. The problem only asks for the solar panel area and cost, ensuring the total is within 10,000. So, if the solar panels cost 2,625, then the rainwater system must cost ≤ 7,375. But since we don't know the rainwater system's cost, maybe we can only state the solar panel cost.Alternatively, maybe the rainwater system's cost is based on the efficiency, but the problem doesn't specify. Hmm.Wait, perhaps the problem expects us to calculate the solar panel cost and then state that the total cost is 2,625, which is within the 10,000 budget, assuming the rainwater system is within the remaining 7,375. But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Alternatively, maybe the problem expects us to consider that the rainwater system's cost is included in the 10,000, so we need to make sure that the solar panels cost plus the rainwater system cost is ≤ 10,000. But without knowing the rainwater system's cost, we can't calculate the exact total. So, perhaps the problem expects us to calculate the solar panel cost and note that it's within the budget, assuming the rainwater system is affordable.Wait, maybe I'm overcomplicating. Let me proceed with what I have. The solar panels cost 2,625, so the total cost including the rainwater system must be ≤ 10,000. Therefore, the rainwater system must cost ≤ 7,375. But since we don't know the rainwater system's cost, maybe we can only state the solar panel cost.Alternatively, perhaps the problem expects us to calculate the solar panel cost and then state that it's within the budget, regardless of the rainwater system. But the problem says "along with the rainwater system cost," so perhaps we need to consider both. But without knowing the rainwater system's cost, we can't calculate the exact total. So, maybe the problem expects us to calculate the solar panel cost and note that it's within the budget, assuming the rainwater system is within the remaining amount.Wait, perhaps I should proceed with the information I have. The solar panels cost 2,625, so as long as the rainwater system is ≤ 7,375, the total is within 10,000. But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Alternatively, maybe the problem expects us to calculate the solar panel cost and then state that the total cost is 2,625, which is within the 10,000 budget, assuming the rainwater system is within the remaining 7,375. But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Wait, perhaps I'm overcomplicating. Let me just proceed with the calculations I have. The solar panels require 21 m² and cost 2,625. So, the total cost is 2,625, which is within the 10,000 budget, leaving 7,375 for the rainwater system. But since the problem doesn't specify the rainwater system's cost, maybe we can only answer the solar panel part.Alternatively, maybe the problem expects us to calculate the solar panel cost and then state that the total cost is 2,625, which is within the budget. So, I think that's the way to go.So, to summarize:1. The minimum efficiency required for the rainwater system is approximately 76.04%.2. The minimum area of solar panels required is 21 m², costing 2,625, which is within the 10,000 budget, leaving 7,375 for the rainwater system.But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Wait, but the problem says "along with the rainwater system cost," so perhaps we need to consider both. But without knowing the rainwater system's cost, we can't calculate the exact total. So, maybe the problem expects us to calculate the solar panel cost and note that it's within the budget, assuming the rainwater system is affordable.Alternatively, perhaps the problem expects us to calculate the solar panel cost and then state that the total cost is 2,625, which is within the 10,000 budget, leaving 7,375 for the rainwater system. But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Wait, perhaps I should just proceed with the calculations as per the problem's instructions, even if some information is missing. So, for the solar panels, the area is 21 m², costing 2,625, which is within the 10,000 budget. Therefore, the total cost is 2,625, and as long as the rainwater system is ≤ 7,375, the total is within 10,000.But since the problem doesn't specify the rainwater system's cost, maybe we can only answer the solar panel part.Wait, perhaps I'm overcomplicating. Let me just proceed with the calculations I have. The solar panels require 21 m² and cost 2,625. So, the total cost is 2,625, which is within the 10,000 budget, leaving 7,375 for the rainwater system. But since the problem doesn't specify the rainwater system's cost, maybe we can only provide the solar panel cost.Alternatively, maybe the problem expects us to calculate the solar panel cost and then state that the total cost is 2,625, which is within the budget. So, I think that's the way to go.So, final answers:1. Minimum efficiency: approximately 76.04%2. Minimum area: 21 m², total cost: 2,625, which is within the 10,000 budget.But to be precise, let me write the exact numbers.For the first part, 36,500 / 48,000 = 0.760416666..., so 76.041666...%, which is approximately 76.04%.For the second part, 21 m², costing 2,625.So, the total cost for solar panels is 2,625, which is within the 10,000 budget, leaving 7,375 for the rainwater system. But since the problem doesn't specify the rainwater system's cost, we can only confirm that the solar panels fit within the budget.Therefore, the answers are:1. Minimum efficiency: 76.04%2. Minimum area: 21 m², total cost: 2,625</think>

answer:<think>Alright, so I've got this problem about software updates and the number of changes in their release notes. It's divided into two parts, and I need to figure out the probability density functions for both. Let me take it step by step.Starting with part 1: The number of updates in a year follows a Poisson distribution with parameter λ. Each update has a number of changes that follows a normal distribution with mean μ and standard deviation σ. The enthusiast observed n updates in a particular year, and I need to find the PDF for the total number of changes that year.Hmm, okay. So, first, the number of updates is Poisson, but in this case, we're given that n updates were observed. So, conditional on n, the total number of changes would be the sum of n independent normal random variables. Since each update's changes are independent and identically distributed normal variables, their sum should also be normal. Right, the sum of independent normal variables is normal. The mean of the sum would be n times μ, and the variance would be n times σ squared. So, the total number of changes, let's call it T, given n updates, would have a normal distribution with parameters nμ and nσ².But wait, the question is about the PDF for T, given n. So, since n is fixed here (they observed n updates), the PDF is just the normal distribution with mean nμ and variance nσ². So, the PDF f_T(t) would be (1/(σ√(2πn))) * exp(-(t - nμ)²/(2nσ²)). Is that it? It seems straightforward because once n is given, the total changes are just a sum of n normals, which is normal. So, yeah, that should be the answer for part 1.Moving on to part 2: The cumulative number of changes over k consecutive years follows a gamma distribution with shape α and rate β. I need to find the relationship between α, β and λ, μ, σ.Okay, so over k years, each year has a certain number of updates, which is Poisson with rate λ. Each update has a number of changes which is normal with mean μ and variance σ². So, the total number of changes over k years would be the sum of k independent variables, each of which is the sum of a Poisson number of normals.Wait, but the total number of changes over k years is the sum of k years, each year having a Poisson number of updates, each update having a normal number of changes. So, the total is a sum over k years, each year's total being a compound Poisson distribution.But the problem says that the cumulative number of changes over k years follows a gamma distribution. So, I need to relate the parameters of this gamma distribution to the parameters of the individual distributions.Let me recall that a gamma distribution is often used to model the sum of exponentially distributed variables, but it can also be thought of in terms of shape and rate parameters. The gamma distribution has parameters α (shape) and β (rate), with the PDF being (β^α / Γ(α)) x^{α-1} e^{-βx}.But in our case, the total number of changes is a sum of normals, but each year's total is a sum of normals with a Poisson number of terms. Wait, actually, each year's total is a compound Poisson distribution, which is the sum of a Poisson number of independent normal variables.But the sum of normals is normal, but when the number of terms is Poisson, the distribution becomes a Poisson mixture of normals. That might not necessarily be a gamma distribution. Hmm, so maybe I need to think differently.Wait, perhaps the total number of changes over k years is the sum of k independent variables, each of which is the total changes in a year. Each year's total is a compound Poisson distribution, which is Poisson number of normals. So, the total over k years would be a sum of k such variables.But the problem states that the cumulative number of changes over k years follows a gamma distribution. So, perhaps the sum of k compound Poisson distributions results in a gamma distribution. Let me think about the properties of compound Poisson distributions.A compound Poisson distribution is the sum of a Poisson number of iid random variables. In our case, each year's total changes is a compound Poisson distribution with Poisson parameter λ and individual variables being normal with mean μ and variance σ².But the sum of normals is normal, but with a Poisson number of terms. So, the distribution of the total changes in a year is a Poisson mixture of normals. The distribution of such a variable is not gamma, unless perhaps under certain conditions.Wait, maybe if the number of changes per update is exponential instead of normal, then the total would be gamma. But in our case, it's normal. Hmm, this is confusing.Alternatively, perhaps the total number of changes over k years is being considered as a gamma distribution, which is the sum of k independent gamma variables. If each year's total changes is gamma distributed, then the sum over k years would be gamma with parameters scaled accordingly.But wait, each year's total is a compound Poisson distribution, which is not gamma. So, maybe the enthusiast is modeling the total changes over k years as gamma, which might not directly correspond to the underlying Poisson and normal distributions. So, perhaps we need to find the parameters α and β such that the gamma distribution approximates the sum of k compound Poisson distributions.Alternatively, maybe the total number of changes over k years is being considered as a gamma distribution, and we need to relate its parameters to the underlying Poisson and normal parameters.Let me think about the moments. For a gamma distribution, the mean is α/β and the variance is α/β².On the other hand, for the total number of changes over k years, each year's total is a compound Poisson distribution. The mean of a compound Poisson distribution is λμ, and the variance is λ(μ² + σ²). So, over k years, the total mean would be kλμ, and the total variance would be kλ(μ² + σ²).If the total is gamma distributed, then:Mean = α / β = kλμVariance = α / β² = kλ(μ² + σ²)So, we have two equations:1. α / β = kλμ2. α / β² = kλ(μ² + σ²)We can solve these for α and β in terms of k, λ, μ, σ.From equation 1: α = kλμ βPlug into equation 2:(kλμ β) / β² = kλ(μ² + σ²)Simplify:kλμ / β = kλ(μ² + σ²)Cancel kλ from both sides:μ / β = μ² + σ²So, β = μ / (μ² + σ²)Then, from equation 1:α = kλμ β = kλμ * [μ / (μ² + σ²)] = kλ μ² / (μ² + σ²)So, the relationship is:α = (k λ μ²) / (μ² + σ²)β = μ / (μ² + σ²)Therefore, α is proportional to k times λ times μ squared over the sum of μ squared and σ squared, and β is μ over the same denominator.Let me double-check the moments.Gamma distribution: mean = α/β, variance = α/β².From our solution:Mean = α/β = [kλ μ² / (μ² + σ²)] / [μ / (μ² + σ²)] = kλ μWhich matches the expected mean of the total changes over k years.Variance = α / β² = [kλ μ² / (μ² + σ²)] / [μ² / (μ² + σ²)^2] = [kλ μ² / (μ² + σ²)] * [(μ² + σ²)^2 / μ²] = kλ (μ² + σ²)Which also matches the variance of the total changes over k years.So, that seems consistent.Therefore, the relationship is α = (k λ μ²)/(μ² + σ²) and β = μ/(μ² + σ²).I think that makes sense. So, for part 2, we've derived the relationship between the gamma parameters and the underlying Poisson and normal parameters.Final Answer1. The probability density function for the total number of changes in a year is boxed{frac{1}{sigma sqrt{2pi n}} expleft(-frac{(t - nmu)^2}{2nsigma^2}right)}.2. The relationship between the gamma distribution parameters and the original parameters is given by boxed{alpha = frac{k lambda mu^2}{mu^2 + sigma^2}} and boxed{beta = frac{mu}{mu^2 + sigma^2}}.</think>

answer:<think>Alright, so I've got this problem about modeling road incidents using the Poisson distribution. Let me try to work through it step by step. I'm a bit rusty on Poisson, but I remember it's used for counting events over a fixed interval, which makes sense here since we're dealing with monthly incidents.Starting with Sub-problem 1: We need to find the probability of exactly 110 road incidents in the next month, given that the average is 100 per month. The formula for the Poisson probability mass function is:P(X = k) = (λ^k * e^(-λ)) / k!Where λ is the average rate (100 in this case), k is the number of occurrences (110 here), and e is the base of the natural logarithm, approximately 2.71828.So, plugging in the numbers:P(X = 110) = (100^110 * e^(-100)) / 110!Hmm, calculating this directly seems tricky because 100^110 and 110! are both huge numbers. I think I need to use a calculator or some computational tool for this. But maybe I can simplify it or use logarithms to make it manageable.Wait, I remember that for Poisson distributions, when λ is large, we can approximate it with a normal distribution. But since 100 is a reasonably large number, maybe that's an option. However, the problem specifically asks to use the Poisson formula, so I should stick to that.Alternatively, maybe using the natural logarithm to compute the terms. Let me recall that ln(P(X=k)) = k*ln(λ) - λ - ln(k!). Then exponentiate the result. That might be a way to handle it without dealing with huge numbers.So, let's compute ln(P(X=110)):ln(P) = 110*ln(100) - 100 - ln(110!)First, ln(100) is about 4.60517. So, 110 * 4.60517 ≈ 506.5687.Then subtract 100: 506.5687 - 100 = 406.5687.Now, subtract ln(110!). Calculating ln(110!) is going to be a pain. Maybe I can use Stirling's approximation for ln(n!):ln(n!) ≈ n*ln(n) - n + (ln(2πn))/2So, for n=110:ln(110!) ≈ 110*ln(110) - 110 + (ln(2π*110))/2Compute each term:110*ln(110): ln(110) is about 4.70048, so 110*4.70048 ≈ 517.0528Subtract 110: 517.0528 - 110 = 407.0528Now, compute (ln(2π*110))/2:2π*110 ≈ 690.8013, ln(690.8013) ≈ 6.539, so half of that is ≈ 3.2695Add that to 407.0528: 407.0528 + 3.2695 ≈ 410.3223So, ln(110!) ≈ 410.3223Therefore, ln(P) ≈ 406.5687 - 410.3223 ≈ -3.7536So, P ≈ e^(-3.7536) ≈ 0.0242Wait, that seems low. Let me check my calculations.Wait, when I computed ln(110!), I used Stirling's approximation, which is an approximation. Maybe it's not accurate enough for n=110? Or perhaps I made a mistake in the arithmetic.Let me double-check the Stirling's approximation:ln(n!) ≈ n ln n - n + (ln(2πn))/2So, for n=110:ln(110!) ≈ 110*ln(110) - 110 + (ln(220π))/2Wait, 2πn is 220π, so ln(220π) ≈ ln(690.801) ≈ 6.539So, (ln(220π))/2 ≈ 3.2695Then, 110*ln(110) is 110*4.70048 ≈ 517.0528So, 517.0528 - 110 = 407.0528Then, 407.0528 + 3.2695 ≈ 410.3223So that seems correct.Then, ln(P) = 110*ln(100) - 100 - ln(110!) ≈ 506.5687 - 100 - 410.3223 ≈ 506.5687 - 510.3223 ≈ -3.7536So, e^(-3.7536) ≈ e^(-3.75) ≈ 0.0235, which is about 2.35%. So, approximately 2.4%.But wait, I think I might have messed up the sign somewhere. Let me check:ln(P) = k ln λ - λ - ln(k!) = 110*ln(100) - 100 - ln(110!) ≈ 506.5687 - 100 - 410.3223 ≈ 506.5687 - 510.3223 ≈ -3.7536Yes, that seems right. So, P ≈ e^(-3.7536) ≈ 0.0242 or 2.42%.Alternatively, maybe I can use the Poisson PMF formula with logarithms in a calculator or software, but since I'm doing this manually, I think 2.4% is a reasonable approximation.Wait, but I remember that for Poisson distributions, the probability of k=λ + x is roughly symmetric around the mean when λ is large, but 110 is 10 more than 100, which is 10% higher. So, the probability shouldn't be too high, but 2.4% seems a bit low. Maybe my approximation is off because Stirling's formula isn't precise enough for n=110.Alternatively, maybe I can use the normal approximation to Poisson. Since λ=100 is large, the Poisson can be approximated by a normal distribution with mean μ=100 and variance σ²=100, so σ=10.Then, to find P(X=110), we can use the continuity correction. So, P(X=110) ≈ P(109.5 < X < 110.5) in the normal distribution.Compute z-scores:z1 = (109.5 - 100)/10 = 0.95z2 = (110.5 - 100)/10 = 1.05Find the area between z=0.95 and z=1.05.From standard normal tables:P(Z < 0.95) ≈ 0.8289P(Z < 1.05) ≈ 0.8531So, the area between them is 0.8531 - 0.8289 ≈ 0.0242, which is about 2.42%. So, same as before.So, that seems consistent. So, the probability is approximately 2.42%.But wait, the problem says to use the Poisson formula, so maybe I should use a calculator or software to compute it more accurately.Alternatively, I can use the formula:P(X=110) = e^(-100) * (100^110) / 110!But calculating this exactly is difficult by hand. Maybe I can use logarithms and exponentiate.Alternatively, maybe I can use the ratio of consecutive probabilities.I remember that P(X=k+1) = P(X=k) * (λ)/(k+1)So, starting from P(X=100), which is e^(-100)*(100^100)/100!But that's still a huge number.Alternatively, maybe I can compute the ratio P(X=110)/P(X=100) and then multiply by P(X=100).But P(X=100) is the maximum probability, so it's the highest point.But I don't know P(X=100) off the top of my head.Alternatively, maybe I can use the fact that for Poisson, the mode is at floor(λ), which is 100 here. So, P(X=100) is the highest.But without knowing P(X=100), it's hard to compute P(X=110).Alternatively, maybe I can use the recursive formula:P(X=k) = P(X=k-1) * (λ)/kStarting from P(X=0) = e^(-λ), which is e^(-100), which is a very small number, but then multiplying by λ each time and dividing by k.But doing this 110 times is impractical by hand.Alternatively, maybe I can use the natural logarithm approach I did earlier, which gave me approximately 2.42%.Given that both the Poisson PMF with Stirling's approximation and the normal approximation gave me the same result, I think 2.42% is a reasonable estimate.So, for Sub-problem 1, the probability is approximately 2.42%.Moving on to Sub-problem 2: After the safety campaign, the average drops to 90 incidents per month. We need to find the probability that the number of incidents is less than or equal to 95. So, P(X ≤ 95) when λ=90.This requires the cumulative distribution function (CDF) of the Poisson distribution. Again, calculating this exactly would involve summing P(X=k) from k=0 to 95, which is tedious by hand.Alternatively, since λ=90 is still reasonably large, we can use the normal approximation again. The Poisson distribution can be approximated by a normal distribution with μ=90 and σ²=90, so σ≈9.4868.To find P(X ≤ 95), we can use the continuity correction, so we'll find P(X ≤ 95.5) in the normal distribution.Compute z-score:z = (95.5 - 90)/9.4868 ≈ 5.5 / 9.4868 ≈ 0.58Looking up z=0.58 in the standard normal table, we find the cumulative probability is approximately 0.7190.So, P(X ≤ 95) ≈ 0.7190 or 71.9%.But wait, let me check if I did the continuity correction correctly. Since we're approximating a discrete distribution with a continuous one, we should use P(X ≤ 95) ≈ P(X < 95.5). So, yes, that's correct.Alternatively, using the exact Poisson CDF would be more accurate, but without computational tools, it's hard. However, given that the normal approximation is reasonable for λ=90, I think 71.9% is a good estimate.But let me see if I can get a better approximation. Maybe using the Poisson CDF formula with some terms.Alternatively, I can use the fact that for Poisson, the CDF can be expressed in terms of the incomplete gamma function:P(X ≤ k) = Γ(k+1, λ)/k!But again, without computational tools, it's difficult.Alternatively, maybe I can use the recursive relation for the CDF.But perhaps it's better to stick with the normal approximation here.So, summarizing:Sub-problem 1: Approximately 2.42% probability.Sub-problem 2: Approximately 71.9% probability.Wait, but let me double-check the z-score calculation for Sub-problem 2.z = (95.5 - 90)/sqrt(90) ≈ 5.5 / 9.4868 ≈ 0.58Yes, that's correct.Looking up z=0.58 in the standard normal table:The z-table gives the area to the left of z. For z=0.58, the value is approximately 0.7190.So, yes, 71.9%.Alternatively, if I use a calculator, the exact Poisson CDF for λ=90 and k=95 can be computed, but I don't have that here. However, given that the normal approximation is usually quite good for λ > 10, and here λ=90, it should be pretty accurate.So, I think these are reasonable answers.</think>