🧠:<think>Alright, so I need to figure out how to determine ( I(t) ) as a function of time ( t ) using the given modified SIR model. The equations provided are:[ frac{dS}{dt} = -beta frac{S I}{N} ][ frac{dI}{dt} = beta frac{S I}{N} - gamma I ][ frac{dR}{dt} = gamma I ]Given the initial conditions: ( S(0) = 990,000 ), ( I(0) = 10,000 ), ( R(0) = 0 ), and ( N = 1,000,000 ). The parameters are ( beta = 0.3 ) and ( gamma = 0.1 ).First, I remember that the SIR model is a system of differential equations, and solving them analytically can be quite complex. However, I think there's a way to find an expression for ( I(t) ) by manipulating these equations.Looking at the equations, I notice that ( frac{dS}{dt} = -beta frac{S I}{N} ) and ( frac{dI}{dt} = beta frac{S I}{N} - gamma I ). If I add these two equations together, the ( beta frac{S I}{N} ) terms cancel out, giving:[ frac{d}{dt}(S + I) = -gamma I ]But since ( S + I + R = N ) (the total population is constant), we can write ( S = N - I - R ). However, I'm not sure if that helps directly. Maybe I should consider the ratio of ( frac{dI}{dt} ) and ( frac{dS}{dt} ) to eliminate time.Let me try dividing ( frac{dI}{dt} ) by ( frac{dS}{dt} ):[ frac{dI}{dS} = frac{beta frac{S I}{N} - gamma I}{-beta frac{S I}{N}} ]Simplify the numerator:[ beta frac{S I}{N} - gamma I = I left( beta frac{S}{N} - gamma right) ]So,[ frac{dI}{dS} = frac{I left( beta frac{S}{N} - gamma right)}{ - beta frac{S I}{N} } ]Simplify this expression:The ( I ) terms cancel out, and we have:[ frac{dI}{dS} = frac{ beta frac{S}{N} - gamma }{ - beta frac{S}{N} } ]Which simplifies further to:[ frac{dI}{dS} = -1 + frac{gamma}{beta frac{S}{N}} ]So,[ frac{dI}{dS} = -1 + frac{gamma N}{beta S} ]This is a separable differential equation. Let me rewrite it:[ frac{dI}{dS} = -1 + frac{gamma N}{beta S} ]Which can be rearranged as:[ dI = left( -1 + frac{gamma N}{beta S} right) dS ]Integrating both sides:[ int dI = int left( -1 + frac{gamma N}{beta S} right) dS ]So,[ I = -S + frac{gamma N}{beta} ln S + C ]Where ( C ) is the constant of integration. Now, we can use the initial condition to find ( C ). At ( t = 0 ), ( S = 990,000 ) and ( I = 10,000 ).Plugging these into the equation:[ 10,000 = -990,000 + frac{gamma N}{beta} ln 990,000 + C ]Let me compute ( frac{gamma N}{beta} ):Given ( gamma = 0.1 ), ( N = 1,000,000 ), ( beta = 0.3 ):[ frac{gamma N}{beta} = frac{0.1 times 1,000,000}{0.3} = frac{100,000}{0.3} approx 333,333.333 ]So,[ 10,000 = -990,000 + 333,333.333 ln 990,000 + C ]Compute ( ln 990,000 ). Let's see, ( ln 1,000,000 = ln(10^6) = 6 ln 10 approx 6 times 2.302585 = 13.81551 ). Since 990,000 is slightly less than 1,000,000, ( ln 990,000 ) will be slightly less than 13.81551. Let me compute it more accurately.Compute ( ln(990,000) = ln(1,000,000 times 0.99) = ln(1,000,000) + ln(0.99) approx 13.81551 + (-0.01005) = 13.80546 ).So,[ 10,000 = -990,000 + 333,333.333 times 13.80546 + C ]Compute ( 333,333.333 times 13.80546 ):First, 333,333.333 * 10 = 3,333,333.33333,333.333 * 3 = 1,000,000 (approximately)333,333.333 * 0.80546 ≈ 333,333.333 * 0.8 = 266,666.666 and 333,333.333 * 0.00546 ≈ 1,825. So total ≈ 266,666.666 + 1,825 ≈ 268,491.666So total 3,333,333.33 + 1,000,000 + 268,491.666 ≈ 4,601,824.996So,[ 10,000 = -990,000 + 4,601,824.996 + C ]Compute ( -990,000 + 4,601,824.996 = 3,611,824.996 )Thus,[ 10,000 = 3,611,824.996 + C ]So,[ C = 10,000 - 3,611,824.996 = -3,591,824.996 ]Therefore, the equation relating ( I ) and ( S ) is:[ I = -S + 333,333.333 ln S - 3,591,824.996 ]Hmm, that seems a bit messy. Maybe I made a mistake in the integration. Let me double-check.Wait, when I integrated ( int left( -1 + frac{gamma N}{beta S} right) dS ), that should be:[ -S + frac{gamma N}{beta} ln S + C ]Yes, that's correct. So plugging in the numbers, I think that's right. Maybe instead of using approximate values, I should keep it symbolic for now.Alternatively, perhaps I can express ( I ) in terms of ( S ), and then find ( S(t) ) as a function of time.But solving for ( S(t) ) seems difficult because it's a transcendental equation. Maybe another approach is needed.Wait, perhaps I can use the fact that ( frac{dI}{dt} = beta frac{S I}{N} - gamma I ). If I can express ( S ) in terms of ( I ), maybe I can write this as a differential equation in terms of ( I ) only.From the equation ( I = -S + frac{gamma N}{beta} ln S + C ), we can express ( S ) in terms of ( I ), but it's implicit and might not be solvable analytically.Alternatively, maybe I can use the concept of the effective reproduction number or look for the peak time.Wait, the second part asks for the time ( t_{text{peak}} ) when ( I(t) ) is maximum. For that, I know that the maximum occurs when ( frac{dI}{dt} = 0 ). So, setting ( frac{dI}{dt} = 0 ):[ beta frac{S I}{N} - gamma I = 0 ][ beta frac{S}{N} = gamma ][ S = frac{gamma N}{beta} ]So, at the peak, ( S = frac{gamma N}{beta} ). Plugging in the values:[ S = frac{0.1 times 1,000,000}{0.3} = frac{100,000}{0.3} approx 333,333.33 ]So, when ( S = 333,333.33 ), ( I(t) ) is at its maximum.Now, to find the time ( t_{text{peak}} ), I need to solve for ( t ) when ( S(t) = 333,333.33 ). But how?Since we have the equation ( I = -S + frac{gamma N}{beta} ln S + C ), and we know ( C ), we can plug ( S = 333,333.33 ) into this equation to find ( I ) at the peak, but we still need to relate this to time.Alternatively, perhaps I can use the fact that ( frac{dI}{dt} = 0 ) at the peak, and use numerical methods to solve for ( t ). But since this is a theoretical problem, maybe there's an analytical approach.Wait, another thought: the SIR model can sometimes be approximated using the logistic growth model when the susceptible population is large. But I'm not sure if that applies here.Alternatively, perhaps I can use the next-generation matrix approach or look for the final size equation, but that might not directly give me ( I(t) ).Wait, maybe I can use the fact that ( frac{dI}{dt} = I (beta frac{S}{N} - gamma) ). So, the growth rate of ( I ) depends on ( beta frac{S}{N} - gamma ). When ( S = frac{gamma N}{beta} ), the growth rate is zero, which is the peak.But to find ( t_{text{peak}} ), I need to integrate from ( t=0 ) to ( t_{text{peak}} ), with ( S ) decreasing from 990,000 to 333,333.33.Wait, perhaps I can write ( frac{dI}{dt} = I (beta frac{S}{N} - gamma) ), and since ( frac{dS}{dt} = -beta frac{S I}{N} ), maybe I can write ( frac{dI}{dS} = frac{beta frac{S}{N} - gamma}{ -beta frac{S}{N} } ), which we did earlier.But that leads us back to the same equation. Maybe instead, I can write ( frac{dI}{dt} = I (beta frac{S}{N} - gamma) ), and since ( S = N - I - R ), but ( R ) is negligible initially, so maybe approximate ( S approx N - I ). But that might not be accurate.Alternatively, perhaps I can use the fact that ( frac{dI}{dt} = beta frac{S I}{N} - gamma I ), and write this as:[ frac{dI}{dt} = I left( beta frac{S}{N} - gamma right) ]Let me denote ( beta frac{S}{N} - gamma = r(t) ), the instantaneous growth rate. So,[ frac{dI}{dt} = r(t) I ]This is a linear differential equation, and its solution is:[ I(t) = I(0) expleft( int_0^t r(tau) dtau right) ]But ( r(t) = beta frac{S(t)}{N} - gamma ), and ( S(t) ) is decreasing as people get infected. So, to find ( I(t) ), I need to know ( S(t) ), which is another differential equation.This seems like a system that might not have an analytical solution, so perhaps I need to use numerical methods. But since this is a theoretical problem, maybe there's an approximation or a way to express ( I(t) ) implicitly.Wait, another approach: using the fact that ( S(t) + I(t) + R(t) = N ), and ( R(t) = gamma int_0^t I(tau) dtau ). So,[ S(t) = N - I(t) - gamma int_0^t I(tau) dtau ]But substituting this into the equation for ( frac{dI}{dt} ) gives:[ frac{dI}{dt} = beta frac{(N - I - gamma int_0^t I(tau) dtau) I}{N} - gamma I ]This is an integro-differential equation, which is more complicated. Maybe I can approximate it by assuming that ( R(t) ) is small initially, so ( S(t) approx N - I(t) ). Then,[ frac{dI}{dt} approx beta frac{(N - I) I}{N} - gamma I ]This simplifies to:[ frac{dI}{dt} = left( beta frac{N - I}{N} - gamma right) I ][ frac{dI}{dt} = left( beta - beta frac{I}{N} - gamma right) I ][ frac{dI}{dt} = (beta - gamma) I - beta frac{I^2}{N} ]This is a Bernoulli equation, which can be linearized by substituting ( u = 1/I ). Let me try that.Let ( u = 1/I ), then ( frac{du}{dt} = -frac{1}{I^2} frac{dI}{dt} ).Substituting into the equation:[ frac{du}{dt} = -frac{1}{I^2} [ (beta - gamma) I - beta frac{I^2}{N} ] ][ frac{du}{dt} = -frac{beta - gamma}{I} + frac{beta}{N} ]So,[ frac{du}{dt} = -(beta - gamma) u + frac{beta}{N} ]This is a linear differential equation in ( u ). The integrating factor is ( e^{int (beta - gamma) dt} = e^{(beta - gamma)t} ).Multiplying both sides by the integrating factor:[ e^{(beta - gamma)t} frac{du}{dt} + (beta - gamma) e^{(beta - gamma)t} u = frac{beta}{N} e^{(beta - gamma)t} ]The left side is the derivative of ( u e^{(beta - gamma)t} ):[ frac{d}{dt} left( u e^{(beta - gamma)t} right ) = frac{beta}{N} e^{(beta - gamma)t} ]Integrate both sides:[ u e^{(beta - gamma)t} = frac{beta}{N} int e^{(beta - gamma)t} dt + C ]Compute the integral:[ int e^{(beta - gamma)t} dt = frac{e^{(beta - gamma)t}}{beta - gamma} + C ]So,[ u e^{(beta - gamma)t} = frac{beta}{N} cdot frac{e^{(beta - gamma)t}}{beta - gamma} + C ]Divide both sides by ( e^{(beta - gamma)t} ):[ u = frac{beta}{N (beta - gamma)} + C e^{-(beta - gamma)t} ]Recall that ( u = 1/I ), so:[ frac{1}{I} = frac{beta}{N (beta - gamma)} + C e^{-(beta - gamma)t} ]Now, apply the initial condition ( I(0) = 10,000 ):[ frac{1}{10,000} = frac{beta}{N (beta - gamma)} + C ]Solve for ( C ):[ C = frac{1}{10,000} - frac{beta}{N (beta - gamma)} ]Plugging in the values:( beta = 0.3 ), ( gamma = 0.1 ), ( N = 1,000,000 ):First, compute ( beta - gamma = 0.3 - 0.1 = 0.2 ).Then,[ frac{beta}{N (beta - gamma)} = frac{0.3}{1,000,000 times 0.2} = frac{0.3}{200,000} = 0.0000015 ]So,[ C = frac{1}{10,000} - 0.0000015 = 0.0001 - 0.0000015 = 0.0000985 ]Thus, the expression for ( u ) is:[ frac{1}{I} = 0.0000015 + 0.0000985 e^{-0.2 t} ]Therefore, solving for ( I ):[ I(t) = frac{1}{0.0000015 + 0.0000985 e^{-0.2 t}} ]Simplify the denominator:Factor out 0.0000015:[ 0.0000015 (1 + frac{0.0000985}{0.0000015} e^{-0.2 t}) ]Compute ( frac{0.0000985}{0.0000015} = frac{985}{15} approx 65.6667 )So,[ I(t) = frac{1}{0.0000015 (1 + 65.6667 e^{-0.2 t})} ][ I(t) = frac{1}{0.0000015} cdot frac{1}{1 + 65.6667 e^{-0.2 t}} ][ I(t) = frac{1,000,000}{1 + 65.6667 e^{-0.2 t}} ]Wait, because ( 1/0.0000015 = 1,000,000 / 1.5 approx 666,666.6667 ). Wait, let me compute that correctly.Wait, ( 1 / 0.0000015 = 1 / (1.5 times 10^{-6}) = (1 / 1.5) times 10^6 approx 0.6666667 times 10^6 = 666,666.6667 ).So,[ I(t) = 666,666.6667 cdot frac{1}{1 + 65.6667 e^{-0.2 t}} ]But wait, that doesn't seem right because at ( t = 0 ), ( I(0) ) should be 10,000. Let me check:At ( t = 0 ):[ I(0) = 666,666.6667 / (1 + 65.6667) = 666,666.6667 / 66.6667 ≈ 10,000 ]Yes, that's correct. So, the expression is:[ I(t) = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]Alternatively, to make it cleaner, we can write:[ I(t) = frac{2}{3} times 10^6 cdot frac{1}{1 + frac{200}{3} e^{-0.2 t}} ]But perhaps it's better to keep it in decimal form for clarity.So, summarizing, the expression for ( I(t) ) is:[ I(t) = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]This is the solution to part 1.For part 2, we need to find ( t_{text{peak}} ), the time when ( I(t) ) is maximum. As established earlier, this occurs when ( S(t) = frac{gamma N}{beta} = 333,333.33 ).But how do we find ( t ) when ( S(t) = 333,333.33 )?We can use the relationship between ( S ) and ( I ) that we derived earlier:[ I = -S + frac{gamma N}{beta} ln S + C ]We have ( C = -3,591,824.996 ), so plugging in ( S = 333,333.33 ):[ I = -333,333.33 + 333,333.333 ln(333,333.33) - 3,591,824.996 ]Compute ( ln(333,333.33) ):( ln(333,333.33) = ln(1,000,000 / 3) = ln(1,000,000) - ln(3) ≈ 13.81551 - 1.098612 ≈ 12.7169 )So,[ I ≈ -333,333.33 + 333,333.333 times 12.7169 - 3,591,824.996 ]Compute ( 333,333.333 times 12.7169 ≈ 333,333.333 times 12 = 4,000,000 ) plus ( 333,333.333 times 0.7169 ≈ 238,888.89 ). So total ≈ 4,238,888.89.Thus,[ I ≈ -333,333.33 + 4,238,888.89 - 3,591,824.996 ≈ (-333,333.33 - 3,591,824.996) + 4,238,888.89 ≈ -3,925,158.326 + 4,238,888.89 ≈ 313,730.564 ]So, at the peak, ( I ≈ 313,730.56 ). Now, to find ( t ) when ( I(t) = 313,730.56 ), we can use the expression for ( I(t) ):[ 313,730.56 = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]Solve for ( t ):Multiply both sides by denominator:[ 313,730.56 (1 + 65.6667 e^{-0.2 t}) = 666,666.6667 ]Divide both sides by 313,730.56:[ 1 + 65.6667 e^{-0.2 t} = frac{666,666.6667}{313,730.56} ≈ 2.126 ]Subtract 1:[ 65.6667 e^{-0.2 t} ≈ 1.126 ]Divide both sides by 65.6667:[ e^{-0.2 t} ≈ frac{1.126}{65.6667} ≈ 0.01714 ]Take natural logarithm:[ -0.2 t ≈ ln(0.01714) ≈ -4.055 ]So,[ t ≈ frac{4.055}{0.2} ≈ 20.275 ]Therefore, the time at which the number of infectious individuals reaches its maximum is approximately 20.28 days.But wait, let me double-check the calculations because I approximated some steps.Starting from:[ I(t) = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]Set ( I(t) = 313,730.56 ):[ 313,730.56 = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]Multiply both sides by denominator:[ 313,730.56 (1 + 65.6667 e^{-0.2 t}) = 666,666.6667 ]Divide both sides by 313,730.56:[ 1 + 65.6667 e^{-0.2 t} = frac{666,666.6667}{313,730.56} ≈ 2.126 ]So,[ 65.6667 e^{-0.2 t} = 2.126 - 1 = 1.126 ]Thus,[ e^{-0.2 t} = frac{1.126}{65.6667} ≈ 0.01714 ]Taking natural log:[ -0.2 t = ln(0.01714) ≈ -4.055 ]So,[ t ≈ frac{4.055}{0.2} ≈ 20.275 ]Yes, that seems consistent. So, approximately 20.28 days.Alternatively, since the model is approximate, maybe we can express it more precisely.But given the approximations made earlier, 20.28 days is a reasonable estimate.So, to summarize:1. The function ( I(t) ) is given by:[ I(t) = frac{666,666.6667}{1 + 65.6667 e^{-0.2 t}} ]2. The time at which ( I(t) ) reaches its maximum is approximately 20.28 days.</think>

🧠:<think>Okay, so I have this problem about a data analyst looking into racial disparities in the criminal justice system. They've got a dataset with arrest records, and they want to do some statistical analysis on it. The problem is split into two parts: one involving Principal Component Analysis (PCA) and another involving a two-sample t-test. Let me tackle each part step by step.Starting with the PCA sub-problem. They want to reduce the dimensionality of the dataset by performing PCA on three features: race (which is encoded numerically), age, and crime severity score. The covariance matrix provided is a 3x3 matrix:[begin{bmatrix}3 & 1.5 & 0.5 1.5 & 2 & 0.3 0.5 & 0.3 & 1end{bmatrix}]I need to calculate the eigenvalues and eigenvectors of this covariance matrix and then determine which principal component captures the most variance.Alright, eigenvalues and eigenvectors. I remember that eigenvalues represent the variance explained by each principal component, and eigenvectors are the directions of these components. The principal component with the largest eigenvalue captures the most variance.To find eigenvalues, I need to solve the characteristic equation, which is given by:[det(C - lambda I) = 0]Where ( C ) is the covariance matrix, ( lambda ) represents the eigenvalues, and ( I ) is the identity matrix.So, let's set up the equation:[detleft( begin{bmatrix}3 - lambda & 1.5 & 0.5 1.5 & 2 - lambda & 0.3 0.5 & 0.3 & 1 - lambdaend{bmatrix} right) = 0]Calculating the determinant of a 3x3 matrix can be a bit tedious, but let's proceed step by step.The determinant of a 3x3 matrix:[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)]Where the matrix is:[begin{bmatrix}a & b & c d & e & f g & h & iend{bmatrix}]Applying this to our matrix:First row: ( a = 3 - lambda ), ( b = 1.5 ), ( c = 0.5 )Second row: ( d = 1.5 ), ( e = 2 - lambda ), ( f = 0.3 )Third row: ( g = 0.5 ), ( h = 0.3 ), ( i = 1 - lambda )So, determinant:( (3 - lambda)[(2 - lambda)(1 - lambda) - (0.3)(0.3)] - 1.5[(1.5)(1 - lambda) - (0.3)(0.5)] + 0.5[(1.5)(0.3) - (2 - lambda)(0.5)] )Let me compute each part step by step.First part: ( (3 - lambda)[(2 - lambda)(1 - lambda) - 0.09] )Compute ( (2 - lambda)(1 - lambda) ):( 2*1 - 2lambda - lambda*1 + lambda^2 = 2 - 3lambda + lambda^2 )Subtract 0.09: ( 2 - 3lambda + lambda^2 - 0.09 = 1.91 - 3lambda + lambda^2 )So first part: ( (3 - lambda)(1.91 - 3lambda + lambda^2) )Second part: ( -1.5[(1.5)(1 - lambda) - 0.15] )Compute inside the brackets: ( 1.5 - 1.5lambda - 0.15 = 1.35 - 1.5lambda )Multiply by -1.5: ( -1.5*(1.35 - 1.5lambda) = -2.025 + 2.25lambda )Third part: ( 0.5[(0.45) - (1 - 0.5lambda)] )Wait, let me compute the inside first: ( (1.5)(0.3) = 0.45 ) and ( (2 - lambda)(0.5) = 1 - 0.5lambda )So, inside the brackets: ( 0.45 - (1 - 0.5lambda) = 0.45 - 1 + 0.5lambda = -0.55 + 0.5lambda )Multiply by 0.5: ( 0.5*(-0.55 + 0.5lambda) = -0.275 + 0.25lambda )Now, putting all three parts together:First part: ( (3 - lambda)(lambda^2 - 3lambda + 1.91) )Second part: ( -2.025 + 2.25lambda )Third part: ( -0.275 + 0.25lambda )So, the entire determinant equation is:( (3 - lambda)(lambda^2 - 3lambda + 1.91) - 2.025 + 2.25lambda - 0.275 + 0.25lambda = 0 )Simplify the constants and lambda terms:-2.025 - 0.275 = -2.32.25λ + 0.25λ = 2.5λSo, the equation becomes:( (3 - lambda)(lambda^2 - 3lambda + 1.91) - 2.3 + 2.5lambda = 0 )Now, let's expand ( (3 - lambda)(lambda^2 - 3lambda + 1.91) ):Multiply term by term:3*(λ² - 3λ + 1.91) = 3λ² - 9λ + 5.73-λ*(λ² - 3λ + 1.91) = -λ³ + 3λ² - 1.91λSo, combining these:3λ² - 9λ + 5.73 - λ³ + 3λ² - 1.91λCombine like terms:-λ³ + (3λ² + 3λ²) + (-9λ - 1.91λ) + 5.73Which is:-λ³ + 6λ² - 10.91λ + 5.73Now, plug this back into the determinant equation:-λ³ + 6λ² - 10.91λ + 5.73 - 2.3 + 2.5λ = 0Combine constants and lambda terms:5.73 - 2.3 = 3.43-10.91λ + 2.5λ = -8.41λSo, the equation becomes:-λ³ + 6λ² - 8.41λ + 3.43 = 0Multiply both sides by -1 to make it easier:λ³ - 6λ² + 8.41λ - 3.43 = 0Now, we have a cubic equation: λ³ - 6λ² + 8.41λ - 3.43 = 0Solving cubic equations can be tricky. Maybe I can try rational roots. The possible rational roots are factors of 3.43 over factors of 1, so ±1, ±3.43, etc. Let me test λ=1:1 - 6 + 8.41 - 3.43 = 1 -6= -5 +8.41=3.41 -3.43≈-0.02. Close to zero, but not exactly. Maybe λ≈1 is a root.Alternatively, perhaps λ=3. Let's test λ=3:27 - 54 + 25.23 -3.43 = 27-54=-27 +25.23=-1.77 -3.43≈-5.2. Not zero.How about λ=2:8 -24 +16.82 -3.43=8-24=-16 +16.82=0.82 -3.43≈-2.61. Not zero.Hmm, maybe λ=0.5:0.125 - 1.5 + 4.205 -3.43≈0.125-1.5=-1.375 +4.205=2.83 -3.43≈-0.6. Not zero.Alternatively, maybe λ≈1. Let's try λ=1. Let's compute f(1)=1 -6 +8.41 -3.43= (1-6)= -5 +8.41=3.41 -3.43≈-0.02. So, very close to zero. So, λ≈1 is a root.Therefore, we can factor out (λ - 1). Let's perform polynomial division or use synthetic division.Using synthetic division for cubic equation:Coefficients: 1 | -6 | 8.41 | -3.43Divide by (λ -1):Bring down 1.Multiply by 1: 1.Add to next coefficient: -6 +1= -5.Multiply by1: -5.Add to next coefficient:8.41 + (-5)=3.41.Multiply by1:3.41.Add to last coefficient: -3.43 +3.41≈-0.02. Which is close to zero, as expected.So, the cubic factors as (λ -1)(λ² -5λ +3.41)≈0So, the quadratic equation is λ² -5λ +3.41=0Using quadratic formula:λ = [5 ± sqrt(25 - 4*1*3.41)] /2Compute discriminant: 25 -13.64=11.36sqrt(11.36)≈3.37So, λ≈(5 ±3.37)/2Thus, λ≈(5 +3.37)/2≈8.37/2≈4.185And λ≈(5 -3.37)/2≈1.63/2≈0.815So, the eigenvalues are approximately 1, 4.185, and 0.815.Wait, but let me check the calculations again because the determinant equation after expanding was:-λ³ +6λ² -8.41λ +3.43=0But when I multiplied by -1, it became λ³ -6λ² +8.41λ -3.43=0Then, testing λ=1 gave f(1)=1 -6 +8.41 -3.43≈-0.02, so approximately 1 is a root.Then, factoring out (λ -1), we get quadratic λ² -5λ +3.41.So, the eigenvalues are approximately 1, (5 + sqrt(11.36))/2≈4.185, and (5 - sqrt(11.36))/2≈0.815So, eigenvalues are approximately 4.185, 1, and 0.815.Wait, but hold on, the trace of the covariance matrix is 3 + 2 +1=6, which should be equal to the sum of eigenvalues. Let's check: 4.185 +1 +0.815≈6, which is correct.So, the eigenvalues are approximately 4.185, 1, and 0.815.Therefore, the principal components are ordered by their eigenvalues. The first principal component corresponds to the largest eigenvalue, which is approximately 4.185.So, the first principal component captures the most variance.Now, to find the eigenvectors corresponding to each eigenvalue.Starting with the largest eigenvalue, λ≈4.185.We need to solve (C - λI)v =0.So, the matrix C - λI is:[begin{bmatrix}3 -4.185 & 1.5 & 0.5 1.5 & 2 -4.185 & 0.3 0.5 & 0.3 & 1 -4.185end{bmatrix}]Which simplifies to:[begin{bmatrix}-1.185 & 1.5 & 0.5 1.5 & -2.185 & 0.3 0.5 & 0.3 & -3.185end{bmatrix}]We can write the system of equations:-1.185v1 +1.5v2 +0.5v3 =01.5v1 -2.185v2 +0.3v3 =00.5v1 +0.3v2 -3.185v3 =0We can try to find a non-trivial solution. Let's pick v3 as a free variable and express v1 and v2 in terms of v3.From the first equation:-1.185v1 +1.5v2 +0.5v3 =0 => 1.185v1 =1.5v2 +0.5v3 => v1=(1.5v2 +0.5v3)/1.185Similarly, from the second equation:1.5v1 -2.185v2 +0.3v3 =0Substitute v1 from above:1.5*(1.5v2 +0.5v3)/1.185 -2.185v2 +0.3v3 =0Compute 1.5/1.185≈1.266So, 1.266*(1.5v2 +0.5v3) -2.185v2 +0.3v3=0Compute 1.266*1.5≈1.899 and 1.266*0.5≈0.633So, 1.899v2 +0.633v3 -2.185v2 +0.3v3=0Combine like terms:(1.899 -2.185)v2 + (0.633 +0.3)v3=0(-0.286)v2 +0.933v3=0So, -0.286v2 = -0.933v3 => v2≈(0.933/0.286)v3≈3.26v3So, v2≈3.26v3From the first equation, v1=(1.5v2 +0.5v3)/1.185≈(1.5*3.26v3 +0.5v3)/1.185≈(4.89v3 +0.5v3)/1.185≈5.39v3/1.185≈4.55v3So, v1≈4.55v3, v2≈3.26v3, v3=v3Thus, the eigenvector is proportional to [4.55, 3.26, 1]We can normalize this vector. Let's compute its magnitude:sqrt(4.55² +3.26² +1²)=sqrt(20.7025 +10.6276 +1)=sqrt(32.3301)≈5.686So, normalized eigenvector≈[4.55/5.686, 3.26/5.686, 1/5.686]≈[0.799, 0.573, 0.176]So, approximately [0.8, 0.57, 0.176]Similarly, we can find eigenvectors for the other eigenvalues, but since the question only asks for the principal component that captures the most variance, which is the first one, we can stop here.So, the principal component is the eigenvector corresponding to the largest eigenvalue, which is approximately 4.185, and the eigenvector is approximately [0.8, 0.57, 0.176].Now, moving on to the hypothesis testing sub-problem.The analyst wants to compare the mean crime severity scores between two racial groups, A and B. The scores for group A are {6.2, 5.9, 6.5, 7.0, 6.8} and for group B are {4.1, 4.5, 4.9, 5.0, 4.3}. Perform a two-sample t-test to determine if there's a statistically significant difference between the means at α=0.05.Alright, two-sample t-test. Since the sample sizes are small (n=5 for each group), we need to check if we can assume equal variances or not. If we don't know, we can perform an F-test to check for equality of variances, but sometimes people use the Welch's t-test which doesn't assume equal variances.Alternatively, since the sample sizes are equal, maybe we can assume equal variances. Let me compute the variances.First, compute the means.Group A: 6.2, 5.9, 6.5, 7.0, 6.8Mean of A: (6.2 +5.9 +6.5 +7.0 +6.8)/5 = (6.2+5.9=12.1; 12.1+6.5=18.6; 18.6+7=25.6; 25.6+6.8=32.4)/5=32.4/5=6.48Group B:4.1,4.5,4.9,5.0,4.3Mean of B: (4.1 +4.5 +4.9 +5.0 +4.3)/5=(4.1+4.5=8.6; 8.6+4.9=13.5; 13.5+5=18.5; 18.5+4.3=22.8)/5=22.8/5=4.56So, mean difference is 6.48 -4.56=1.92Now, compute the variances.Variance of A:Each score minus mean: 6.2-6.48=-0.28; 5.9-6.48=-0.58; 6.5-6.48=0.02; 7.0-6.48=0.52; 6.8-6.48=0.32Squares: (-0.28)^2=0.0784; (-0.58)^2=0.3364; (0.02)^2=0.0004; (0.52)^2=0.2704; (0.32)^2=0.1024Sum of squares:0.0784+0.3364=0.4148 +0.0004=0.4152 +0.2704=0.6856 +0.1024=0.788Variance of A: 0.788/(5-1)=0.788/4=0.197Variance of B:Each score minus mean:4.1-4.56=-0.46;4.5-4.56=-0.06;4.9-4.56=0.34;5.0-4.56=0.44;4.3-4.56=-0.26Squares: (-0.46)^2=0.2116; (-0.06)^2=0.0036; (0.34)^2=0.1156; (0.44)^2=0.1936; (-0.26)^2=0.0676Sum of squares:0.2116+0.0036=0.2152 +0.1156=0.3308 +0.1936=0.5244 +0.0676=0.592Variance of B:0.592/(5-1)=0.592/4=0.148So, variances are 0.197 and 0.148. They are not equal, but not extremely different. Since the sample sizes are equal, we can proceed with the pooled variance t-test.Pooled variance formula:s_p² = [(n1 -1)s1² + (n2 -1)s2²]/(n1 +n2 -2)Here, n1=n2=5, s1²=0.197, s2²=0.148So, s_p²=(4*0.197 +4*0.148)/(5+5-2)= (0.788 +0.592)/8=1.38/8=0.1725Pooled standard deviation s_p= sqrt(0.1725)≈0.415Now, the t-statistic is:t = (M1 - M2)/sqrt(s_p²*(1/n1 +1/n2)) = (6.48 -4.56)/sqrt(0.1725*(1/5 +1/5)) =1.92/sqrt(0.1725*(0.2 +0.2))=1.92/sqrt(0.1725*0.4)=1.92/sqrt(0.069)=1.92/0.2627≈7.31Degrees of freedom: n1 +n2 -2=5+5-2=8Now, we need to compare this t-statistic to the critical value from the t-distribution table with df=8 and α=0.05.Since it's a two-tailed test, the critical value is t_(α/2, df)=t_(0.025,8). From the table, t critical≈2.306Our calculated t≈7.31 is much larger than 2.306, so we reject the null hypothesis.Alternatively, we can compute the p-value. With t=7.31 and df=8, the p-value is extremely small, definitely less than 0.05.Therefore, we conclude that there is a statistically significant difference between the mean crime severity scores of the two groups.Wait, but hold on, I assumed equal variances because the sample sizes are equal. But the variances are 0.197 and 0.148. The ratio is about 0.197/0.148≈1.33, which is not too large, so equal variances assumption might be acceptable. However, sometimes people prefer Welch's t-test when variances are unequal, especially with small sample sizes. Let me compute Welch's t-test as well.Welch's t-test formula:t = (M1 - M2)/sqrt(s1²/n1 + s2²/n2) =1.92/sqrt(0.197/5 +0.148/5)=1.92/sqrt(0.0394 +0.0296)=1.92/sqrt(0.069)=1.92/0.2627≈7.31Same t-statistic as before.Degrees of freedom for Welch's test is calculated using the Welch-Satterthwaite equation:df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1)]Plugging in the values:s1²=0.197, n1=5; s2²=0.148, n2=5Compute numerator: (0.197/5 +0.148/5)²=(0.0394 +0.0296)²=(0.069)²=0.004761Denominator: (0.0394²)/(4) + (0.0296²)/(4)= (0.001552)/(4) + (0.000876)/(4)=0.000388 +0.000219≈0.000607So, df=0.004761 /0.000607≈7.85≈7.85So, approximately 7.85 degrees of freedom. We can round down to 7 or use 7.85.Looking up the critical value for t with df≈7.85 and α=0.05 two-tailed. The critical value is approximately between t=2.365 (df=7) and t=2.306 (df=8). Let's say approximately 2.33.Our t-statistic is 7.31, which is still way larger than 2.33, so again, we reject the null hypothesis.Therefore, regardless of whether we use pooled variance or Welch's test, the conclusion is the same: there's a statistically significant difference between the two groups.So, summarizing:1. The eigenvalues are approximately 4.185, 1, and 0.815. The principal component capturing the most variance is the one with eigenvalue≈4.185, and its eigenvector is approximately [0.8, 0.57, 0.176].2. The two-sample t-test shows a statistically significant difference between the mean crime severity scores of groups A and B.Final Answer1. The principal component capturing the most variance is associated with the eigenvalue boxed{4.185}.2. The two-sample t-test results in a statistically significant difference between the groups, so we reject the null hypothesis. The conclusion is boxed{text{Reject } H_0}.</think>

🧠:<think>First, I need to determine the dimensions of the central prayer area that follows the golden ratio. The golden ratio is approximately 1.618, meaning the length is about 1.618 times the width.Let’s denote the width of the prayer area as ( w ) meters. Then, the length would be ( 1.618w ) meters. The area of the prayer area is ( 1.618w^2 ).Next, I'll calculate the area of the remaining space by subtracting the prayer area from the total mosque area. The total area of the mosque is ( 50 times 30 = 1500 ) square meters. So, the remaining area is ( 1500 - 1.618w^2 ).Imam Ahmed wants to divide this remaining area into two equal sections. Each section will have an area of ( frac{1500 - 1.618w^2}{2} ).To find the dimensions of each section, I need to determine the width and length that satisfy this area. Assuming the width of each section is the same as the width of the mosque, which is 30 meters, the length of each section would be ( frac{1500 - 1.618w^2}{60} ).Finally, I'll verify that the dimensions of the prayer area indeed follow the golden ratio by checking if the ratio of length to width is approximately 1.618.</think>

🧠:<think>Alright, so I've got this problem about designing Rocket Raccoon's costume. It's divided into two parts, and I need to figure out both. Let's start with the first one.Problem 1: Scaling the HeadpieceOkay, the headpiece is a sphere with a diameter of 12 inches. The designer wants to scale it down to 3/4 of its original size. I need to find the new diameter and the new surface area.First, scaling down by 3/4. So, if the original diameter is 12 inches, scaling it down by 3/4 would mean multiplying 12 by 3/4. Let me write that down:New diameter = 12 inches * (3/4) = 9 inches.That seems straightforward. So, the new diameter is 9 inches.Now, for the surface area. The surface area of a sphere is given by the formula 4πr², where r is the radius. Since the diameter is 12 inches, the original radius is 6 inches. After scaling, the new diameter is 9 inches, so the new radius is 4.5 inches.Calculating the original surface area:Original Surface Area = 4π*(6)² = 4π*36 = 144π square inches.New Surface Area = 4π*(4.5)² = 4π*20.25 = 81π square inches.Wait, but scaling affects surface area by the square of the scaling factor. So, instead of calculating both, maybe I can just use the scaling factor squared.Scaling factor is 3/4, so the surface area scaling factor is (3/4)² = 9/16.Therefore, new surface area = original surface area * 9/16.Original surface area was 144π, so 144π * 9/16 = (144/16)*9π = 9*9π = 81π. Yep, same result. So, that checks out.So, the new diameter is 9 inches, and the new surface area is 81π square inches.Problem 2: Tail Length and WidthAlright, the tail is composed of the first 10 Fibonacci numbers in inches, starting from 1 inch. I need to find the total length. Then, each segment's width decreases exponentially by a factor of 0.8, starting from 2 inches. I need the total width after all 10 segments.First, let's list the first 10 Fibonacci numbers. Fibonacci sequence starts with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Wait, is that 10 numbers? Let me count: 1 (1st), 1 (2nd), 2 (3rd), 3 (4th), 5 (5th), 8 (6th), 13 (7th), 21 (8th), 34 (9th), 55 (10th). Yep, that's 10.So, the lengths are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 inches.Total length is the sum of these. Let me add them up step by step.1 + 1 = 22 + 2 = 44 + 3 = 77 + 5 = 1212 + 8 = 2020 + 13 = 3333 + 21 = 5454 + 34 = 8888 + 55 = 143 inches.So, the total length of the tail is 143 inches.Now, for the width. Each segment's width decreases exponentially by a factor of 0.8, starting from 2 inches. So, the first segment is 2 inches wide, the second is 2*0.8, the third is 2*(0.8)^2, and so on, up to the 10th segment.This is a geometric series where the first term a = 2, common ratio r = 0.8, and number of terms n = 10.The formula for the sum of a geometric series is S_n = a*(1 - r^n)/(1 - r).Plugging in the values:S_10 = 2*(1 - 0.8^10)/(1 - 0.8)First, calculate 0.8^10. Let me compute that.0.8^1 = 0.80.8^2 = 0.640.8^3 = 0.5120.8^4 = 0.40960.8^5 = 0.327680.8^6 = 0.2621440.8^7 = 0.20971520.8^8 = 0.167772160.8^9 = 0.1342177280.8^10 = 0.1073741824So, 0.8^10 ≈ 0.1073741824Now, compute 1 - 0.1073741824 = 0.8926258176Then, 1 - 0.8 = 0.2So, S_10 = 2*(0.8926258176)/0.2 = 2*(4.463129088) = 8.926258176 inches.Approximately, that's 8.926 inches.But let me verify the calculation step by step.Compute numerator: 1 - 0.8^10 ≈ 1 - 0.1073741824 = 0.8926258176Denominator: 1 - 0.8 = 0.2So, 0.8926258176 / 0.2 = 4.463129088Multiply by 2: 4.463129088 * 2 = 8.926258176Yes, that's correct. So, the total width is approximately 8.926 inches.But since we're dealing with measurements, maybe we can round it to a reasonable decimal place. Let's say 8.93 inches.Alternatively, if we want to be precise, we can write it as 8.926 inches.But let me check if I did everything correctly.First term: 2Second term: 2*0.8 = 1.6Third term: 2*0.8^2 = 1.28Fourth term: 2*0.8^3 = 1.024Fifth term: 2*0.8^4 = 0.8192Sixth term: 2*0.8^5 = 0.65536Seventh term: 2*0.8^6 = 0.524288Eighth term: 2*0.8^7 = 0.4194304Ninth term: 2*0.8^8 = 0.33554432Tenth term: 2*0.8^9 = 0.268435456Now, let's add all these up:2 + 1.6 = 3.63.6 + 1.28 = 4.884.88 + 1.024 = 5.9045.904 + 0.8192 = 6.72326.7232 + 0.65536 = 7.378567.37856 + 0.524288 = 7.9028487.902848 + 0.4194304 = 8.32227848.3222784 + 0.33554432 = 8.657822728.65782272 + 0.268435456 = 8.926258176Yes, exactly the same as before. So, the total width is 8.926258176 inches, which is approximately 8.93 inches.So, summarizing:Total tail length: 143 inches.Total tail width: approximately 8.93 inches.Wait, but the problem says "the total width of the tail after all 10 segments." So, is it the sum of all widths? Yes, because each segment has its own width, so adding them up gives the total width.Alternatively, sometimes people might interpret width as the width at the end, but the problem says "the total width of the tail after all 10 segments," which suggests summing them up. So, I think my approach is correct.Just to make sure, the first segment is 2 inches wide, the next is 1.6, and so on, each subsequent segment being 0.8 times the previous width. So, adding all these widths gives the total width of the tail.Yes, that makes sense.Final Answer1. The new diameter is boxed{9} inches and the new surface area is boxed{81pi} square inches.2. The total length of the tail is boxed{143} inches and the total width is approximately boxed{8.93} inches.</think>