🧠:<think>Alright, so I have this problem where a playwright is using a mathematical model to simulate character interactions in a play. The model uses a graph where each character is a vertex, and the edges represent interactions with weights based on dialogue lines. Part 1: There are 5 characters, A, B, C, D, and E, with lines 10, 20, 30, 40, and 50 respectively. The interaction strength between any two characters is given by the function f(x, y) = (xy)/(x + y). I need to calculate the total interaction strength.First, I need to figure out all the possible interactions between these characters. Since it's a graph, each pair of characters will have an edge connecting them, so I need to compute f(x, y) for each pair and then sum them all up.Let me list all the pairs:- A & B- A & C- A & D- A & E- B & C- B & D- B & E- C & D- C & E- D & EThat's 10 pairs in total.Now, I'll compute f(x, y) for each pair.Starting with A & B:x = 10, y = 20f(A,B) = (10*20)/(10+20) = 200/30 ≈ 6.6667A & C:x = 10, y = 30f(A,C) = (10*30)/(10+30) = 300/40 = 7.5A & D:x = 10, y = 40f(A,D) = (10*40)/(10+40) = 400/50 = 8A & E:x = 10, y = 50f(A,E) = (10*50)/(10+50) = 500/60 ≈ 8.3333B & C:x = 20, y = 30f(B,C) = (20*30)/(20+30) = 600/50 = 12B & D:x = 20, y = 40f(B,D) = (20*40)/(20+40) = 800/60 ≈ 13.3333B & E:x = 20, y = 50f(B,E) = (20*50)/(20+50) = 1000/70 ≈ 14.2857C & D:x = 30, y = 40f(C,D) = (30*40)/(30+40) = 1200/70 ≈ 17.1429C & E:x = 30, y = 50f(C,E) = (30*50)/(30+50) = 1500/80 = 18.75D & E:x = 40, y = 50f(D,E) = (40*50)/(40+50) = 2000/90 ≈ 22.2222Now, I need to sum all these values:6.6667 + 7.5 + 8 + 8.3333 + 12 + 13.3333 + 14.2857 + 17.1429 + 18.75 + 22.2222Let me compute step by step:Start with 6.6667 + 7.5 = 14.166714.1667 + 8 = 22.166722.1667 + 8.3333 = 30.530.5 + 12 = 42.542.5 + 13.3333 = 55.833355.8333 + 14.2857 ≈ 70.11970.119 + 17.1429 ≈ 87.261987.2619 + 18.75 ≈ 106.0119106.0119 + 22.2222 ≈ 128.2341So, approximately 128.2341 is the total interaction strength.Wait, let me double-check the calculations because adding decimals can lead to errors.Alternatively, maybe I should compute each fraction exactly and then sum them.Let me try that.f(A,B) = 200/30 = 20/3 ≈ 6.6667f(A,C) = 300/40 = 30/4 = 15/2 = 7.5f(A,D) = 400/50 = 8f(A,E) = 500/60 = 50/6 ≈ 8.3333f(B,C) = 600/50 = 12f(B,D) = 800/60 = 40/3 ≈ 13.3333f(B,E) = 1000/70 = 100/7 ≈ 14.2857f(C,D) = 1200/70 = 120/7 ≈ 17.1429f(C,E) = 1500/80 = 150/8 = 75/4 = 18.75f(D,E) = 2000/90 = 200/9 ≈ 22.2222Now, let's express all fractions with a common denominator to sum them exactly. The denominators are 3, 2, 1, 6, 1, 3, 7, 7, 4, 9.The least common multiple (LCM) of denominators 3, 2, 1, 6, 3, 7, 7, 4, 9.Factors:3: 32: 21: 16: 2, 33: 37: 77: 74: 2²9: 3²So LCM is 2² * 3² * 7 = 4 * 9 * 7 = 252.Convert each fraction to denominator 252:20/3 = (20 * 84)/252 = 1680/25215/2 = (15 * 126)/252 = 1890/2528 = 8/1 = (8 * 252)/252 = 2016/25250/6 = (50 * 42)/252 = 2100/25212 = 12/1 = (12 * 252)/252 = 3024/25240/3 = (40 * 84)/252 = 3360/252100/7 = (100 * 36)/252 = 3600/252120/7 = (120 * 36)/252 = 4320/25275/4 = (75 * 63)/252 = 4725/252200/9 = (200 * 28)/252 = 5600/252Now, sum all numerators:1680 + 1890 + 2016 + 2100 + 3024 + 3360 + 3600 + 4320 + 4725 + 5600Let me compute step by step:Start with 1680 + 1890 = 35703570 + 2016 = 55865586 + 2100 = 76867686 + 3024 = 1071010710 + 3360 = 1407014070 + 3600 = 1767017670 + 4320 = 2199021990 + 4725 = 2671526715 + 5600 = 32315So total numerator is 32315.Therefore, total interaction strength is 32315/252.Simplify this fraction:Divide numerator and denominator by GCD(32315, 252). Let's find GCD.252 divides into 32315 how many times?252 * 128 = 3225632315 - 32256 = 59So GCD(252, 59). 252 ÷ 59 = 4 with remainder 16.GCD(59,16). 59 ÷16=3 rem 11GCD(16,11). 16 ÷11=1 rem5GCD(11,5). 11 ÷5=2 rem1GCD(5,1)=1So GCD is 1. Therefore, 32315/252 is in simplest terms.Convert to decimal: 32315 ÷ 252 ≈ 128.2341So, the exact total is 32315/252 ≈ 128.2341.So, approximately 128.23.Wait, but the problem didn't specify whether to give an exact fraction or a decimal. Since the function f(x,y) is given as a fraction, maybe it's better to present the exact value.But 32315/252 is a bit unwieldy. Let me see if I can simplify it more or perhaps express it as a mixed number.32315 ÷ 252: 252*128=32256, remainder 59. So 128 and 59/252.59/252 can be simplified? 59 is prime, 252=4*63=4*7*9. 59 doesn't divide into 252, so 59/252 is simplest.So, total interaction strength is 128 59/252, or approximately 128.23.So, I think either is acceptable, but since the question didn't specify, maybe both.But in the answer, I should probably present it as a fraction or a decimal. Since the initial function is a fraction, perhaps the exact fraction is better.But 32315/252 is correct.Alternatively, maybe I made a mistake in adding the numerators. Let me double-check:1680 (A,B)+1890 (A,C) = 3570+2016 (A,D) = 5586+2100 (A,E) = 7686+3024 (B,C) = 10710+3360 (B,D) = 14070+3600 (B,E) = 17670+4320 (C,D) = 21990+4725 (C,E) = 26715+5600 (D,E) = 32315Yes, that seems correct.So, 32315/252 ≈ 128.2341.So, I think that's the total interaction strength.Part 2: The playwright adds a new character F who interacts equally with all existing characters. The interaction strength for F with any other character is given by g(x) = sqrt(x), where x is the number of lines spoken by the other character. F speaks 25 lines. Determine the total interaction strength involving F.So, F interacts with A, B, C, D, E. Each interaction is g(x) where x is the lines of the other character.So, for each existing character, compute g(x) = sqrt(x), then sum them up.Given that F speaks 25 lines, but in the function g(x), it's only dependent on the other character's lines, not F's. So, we just need to compute sqrt(10) + sqrt(20) + sqrt(30) + sqrt(40) + sqrt(50).Compute each:sqrt(10) ≈ 3.1623sqrt(20) ≈ 4.4721sqrt(30) ≈ 5.4772sqrt(40) ≈ 6.3246sqrt(50) ≈ 7.0711Sum them:3.1623 + 4.4721 = 7.63447.6344 + 5.4772 = 13.111613.1116 + 6.3246 = 19.436219.4362 + 7.0711 ≈ 26.5073So, approximately 26.5073.Alternatively, exact form would be sqrt(10) + sqrt(20) + sqrt(30) + sqrt(40) + sqrt(50). But we can factor some terms:sqrt(10) = sqrt(10)sqrt(20) = 2*sqrt(5)sqrt(30) = sqrt(30)sqrt(40) = 2*sqrt(10)sqrt(50) = 5*sqrt(2)So, total interaction strength is sqrt(10) + 2*sqrt(5) + sqrt(30) + 2*sqrt(10) + 5*sqrt(2)Combine like terms:sqrt(10) + 2*sqrt(10) = 3*sqrt(10)So, total is 3*sqrt(10) + 2*sqrt(5) + sqrt(30) + 5*sqrt(2)But I don't think this simplifies further, so the exact value is 3√10 + 2√5 + √30 + 5√2.Alternatively, we can write it as √2(5 + 3√5) + √5(2) + √30, but that might not be simpler.So, either the approximate decimal or the exact expression is acceptable. The problem says "determine the total interaction strength," so both might be acceptable, but since it's a mathematical model, exact form is preferable.But let me check if the function g(x) is applied correctly. The problem says "interaction strength for the new character F with any other character is given by g(x) = sqrt(x), where x is the number of lines spoken by the other character."Yes, so for each existing character, we take their lines, compute sqrt(x), and sum them for F's interactions.So, the total interaction strength involving F is sqrt(10) + sqrt(20) + sqrt(30) + sqrt(40) + sqrt(50).Which is approximately 26.5073.So, summarizing:1. Total interaction strength without F: 32315/252 ≈ 128.23412. Total interaction strength involving F: sqrt(10) + sqrt(20) + sqrt(30) + sqrt(40) + sqrt(50) ≈ 26.5073I think that's it.</think>

🧠:<think>Okay, so I have this problem about a documentary filmmaker who wants to capture the geometric marvels of a cathedral's dome. The dome is cylindrical with an elliptical base. I need to find two things: the volume of the dome and the total surface area, including the base and the curved surface.Starting with the first part: deriving an integral expression for the volume. Hmm, I remember that for volumes of revolution, we can use methods like the disk method or the shell method. But this is an elliptical base, not a circular one. So maybe I need to think about how to extend the disk method to an ellipse.An ellipse has the equation (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a) is the semi-major axis and (b) is the semi-minor axis. If I rotate this ellipse around the z-axis (assuming the height is along the z-axis), it should form an elliptical cylinder or a dome, depending on how it's constructed.Wait, the problem says it's a cylindrical dome. So maybe it's an elliptic cylinder with height (h). So, the volume of a cylinder is usually base area times height. For a circular cylinder, it's (pi r^2 h). For an elliptical cylinder, the base area is (pi a b), so the volume would be (pi a b h). Is that right? But the problem says to derive an integral expression, so maybe I need to set up the integral instead of just using the formula.Alright, let's think about slicing the dome horizontally at a height (z). Each slice would be an ellipse, right? The area of each slice would depend on (z). But wait, if it's a cylinder, the cross-sectional area doesn't change with height. So each horizontal slice is just the same ellipse with area (pi a b). So integrating this from (z = 0) to (z = h) would give the volume.So, the integral expression would be:[V = int_{0}^{h} pi a b , dz]Since (pi a b) is constant with respect to (z), integrating from 0 to (h) just gives (pi a b h), which matches the formula I remembered. So that seems straightforward.Moving on to the second part: the total surface area, including the base and the curved surface. The total surface area would be the sum of the lateral (curved) surface area and the area of the base.First, the base is an ellipse, so its area is (pi a b). Now, the curved surface area. For a cylinder, the lateral surface area is usually (2pi r h), but again, this is an elliptical cylinder, so it's a bit different.I recall that for an elliptic cylinder, the lateral surface area can be found by considering the circumference of the ellipse and multiplying it by the height. But the circumference of an ellipse isn't as straightforward as that of a circle. The exact formula involves an integral or an approximation.Wait, the problem mentions that the curved surface should be considered as a section of an elliptic cylinder. So, maybe I need to derive the lateral surface area using an integral.Let me think about parametrizing the surface. If I consider the ellipse in the xy-plane, parameterized by (x = a cos theta), (y = b sin theta), where (theta) goes from 0 to (2pi). Then, as we move along the height (z) from 0 to (h), each point on the ellipse traces a vertical line. So, the surface can be parametrized by ((theta, z)) where (0 leq theta leq 2pi) and (0 leq z leq h).To find the surface area, I can use the formula for the surface area of a parametric surface:[A = int_{0}^{2pi} int_{0}^{h} left| frac{partial mathbf{r}}{partial theta} times frac{partial mathbf{r}}{partial z} right| dz , dtheta]Where (mathbf{r}(theta, z)) is the parametrization of the surface. Let's write that out.[mathbf{r}(theta, z) = (a cos theta, b sin theta, z)]Compute the partial derivatives:[frac{partial mathbf{r}}{partial theta} = (-a sin theta, b cos theta, 0)][frac{partial mathbf{r}}{partial z} = (0, 0, 1)]Now, take the cross product:[frac{partial mathbf{r}}{partial theta} times frac{partial mathbf{r}}{partial z} = begin{vmatrix}mathbf{i} & mathbf{j} & mathbf{k} -a sin theta & b cos theta & 0 0 & 0 & 1end{vmatrix}= (b cos theta, a sin theta, 0)]The magnitude of this vector is:[sqrt{(b cos theta)^2 + (a sin theta)^2} = sqrt{b^2 cos^2 theta + a^2 sin^2 theta}]So, the surface area integral becomes:[A = int_{0}^{2pi} int_{0}^{h} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} , dz , dtheta]Since the integrand doesn't depend on (z), we can integrate with respect to (z) first:[A = int_{0}^{2pi} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} cdot h , dtheta = h int_{0}^{2pi} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} , dtheta]Hmm, this integral doesn't have an elementary antiderivative, right? It's related to the circumference of an ellipse, which is known to be an elliptic integral. So, I think this is as far as we can go analytically. Therefore, the lateral surface area is:[A_{text{lateral}} = h int_{0}^{2pi} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} , dtheta]And the total surface area would be the lateral surface area plus the area of the base:[A_{text{total}} = A_{text{lateral}} + pi a b = h int_{0}^{2pi} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} , dtheta + pi a b]But wait, the problem mentions the dome, which is a cylindrical dome. Is the top open or closed? If it's a dome, maybe it's only the curved surface and the base, but no top. So, in that case, the total surface area would indeed be the lateral surface plus the base. If it were a full cylinder, we would have two bases, but since it's a dome, probably just one base.So, I think that's the expression. It can't be simplified further without approximation, so this integral expression is the answer.Let me recap:1. Volume: Integral of the base area over height, which is straightforward since the cross-section is constant. So, (V = pi a b h), but expressed as an integral, it's (int_{0}^{h} pi a b , dz).2. Surface Area: The lateral surface area requires a double integral over the parameterization, which simplifies to a single integral involving the square root of (b^2 cos^2 theta + a^2 sin^2 theta), multiplied by height, plus the base area.I think that's it. I don't see any mistakes in the reasoning. The key was recognizing that the volume is straightforward because the cross-sectional area doesn't change, whereas the surface area requires a more involved parametrization and integral.Final Answer1. The volume of the cylindrical dome is given by the integral expression (boxed{int_{0}^{h} pi a b , dz}).2. The total surface area of the dome is given by the expression (boxed{pi a b + h int_{0}^{2pi} sqrt{b^2 cos^2 theta + a^2 sin^2 theta} , dtheta}).</think>

🧠:<think>Okay, so I have this problem about modeling a veteran's discipline and determination using a mathematical function. It's a bit abstract, but I think I can handle it. Let me start by reading the problem again and breaking it down.The function given is D(t) = A e^{-kt} sin(ωt + φ) + B. So, this represents the discipline over time, with t being the years since leaving active duty. The constants are A, k, ω, φ, and B, each with their own meanings: A is the initial discipline amplitude, k is the decay rate, ω is the frequency of oscillation, φ is the phase shift, and B is the baseline discipline level.The first part of the problem says that the maximum discipline is achieved at t = π/(2ω), and D(π/(2ω)) = 150. Also, B is given as 100. I need to find the relationship between A, k, ω, and φ.Alright, so let's think about this. The function D(t) is a combination of an exponential decay term and a sine function, plus a constant baseline. The maximum value of the sine function is 1, so the maximum of D(t) should be when sin(ωt + φ) = 1. That would make the term A e^{-kt} sin(...) equal to A e^{-kt}, and then adding B gives the maximum D(t).Given that the maximum occurs at t = π/(2ω), let's plug that into the function.So, D(π/(2ω)) = A e^{-k*(π/(2ω))} sin(ω*(π/(2ω)) + φ) + B.Simplify the sine term: ω*(π/(2ω)) is π/2, so sin(π/2 + φ). Hmm, sin(π/2 + φ) is equal to cos(φ), right? Because sin(π/2 + x) = cos(x). So, that simplifies to cos(φ).So, D(π/(2ω)) = A e^{-kπ/(2ω)} cos(φ) + B.We know that D(π/(2ω)) is 150, and B is 100. So, plugging in:150 = A e^{-kπ/(2ω)} cos(φ) + 100.Subtract 100 from both sides:50 = A e^{-kπ/(2ω)} cos(φ).So, that's the relationship between A, k, ω, and φ. It's 50 equals A times e to the power of (-kπ/(2ω)) times cos(φ). So, that's the first part done.Now, moving on to the second part. The student hypothesizes that the determination R(t) is the integral of the discipline over time, so R(t) = integral from 0 to t of D(x) dx.We need to find the general form of R(t) in terms of A, k, ω, φ, and B.So, let's write out the integral:R(t) = ∫₀ᵗ [A e^{-kx} sin(ωx + φ) + B] dx.We can split this integral into two parts:R(t) = ∫₀ᵗ A e^{-kx} sin(ωx + φ) dx + ∫₀ᵗ B dx.The second integral is straightforward: ∫₀ᵗ B dx = Bt.The first integral is a bit more complicated. It's the integral of A e^{-kx} sin(ωx + φ) dx from 0 to t. I remember that integrating e^{ax} sin(bx + c) can be done using integration by parts or using a standard formula.Let me recall the formula for ∫ e^{ax} sin(bx + c) dx. I think it's something like e^{ax} [a sin(bx + c) - b cos(bx + c)] / (a² + b²) + C. Let me verify that.Let me set u = e^{ax}, dv = sin(bx + c) dx.Then du = a e^{ax} dx, and v = - (1/b) cos(bx + c).So, integration by parts gives:∫ e^{ax} sin(bx + c) dx = - (e^{ax}/b) cos(bx + c) + (a/b) ∫ e^{ax} cos(bx + c) dx.Now, we need to integrate e^{ax} cos(bx + c). Let's do integration by parts again.Let u = e^{ax}, dv = cos(bx + c) dx.Then du = a e^{ax} dx, and v = (1/b) sin(bx + c).So, ∫ e^{ax} cos(bx + c) dx = (e^{ax}/b) sin(bx + c) - (a/b) ∫ e^{ax} sin(bx + c) dx.Putting it back into the previous equation:∫ e^{ax} sin(bx + c) dx = - (e^{ax}/b) cos(bx + c) + (a/b)[ (e^{ax}/b) sin(bx + c) - (a/b) ∫ e^{ax} sin(bx + c) dx ].Let me write this out:∫ e^{ax} sin(bx + c) dx = - (e^{ax}/b) cos(bx + c) + (a/b²) e^{ax} sin(bx + c) - (a²/b²) ∫ e^{ax} sin(bx + c) dx.Now, let's move the integral term to the left side:∫ e^{ax} sin(bx + c) dx + (a²/b²) ∫ e^{ax} sin(bx + c) dx = - (e^{ax}/b) cos(bx + c) + (a/b²) e^{ax} sin(bx + c).Factor out the integral:[1 + (a²/b²)] ∫ e^{ax} sin(bx + c) dx = e^{ax} [ - (1/b) cos(bx + c) + (a/b²) sin(bx + c) ].So, ∫ e^{ax} sin(bx + c) dx = e^{ax} [ - (1/b) cos(bx + c) + (a/b²) sin(bx + c) ] / [1 + (a²/b²)].Simplify the denominator: 1 + (a²/b²) = (b² + a²)/b².So, flipping that, the integral becomes:e^{ax} [ - (1/b) cos(bx + c) + (a/b²) sin(bx + c) ] * (b²)/(a² + b²).Simplify numerator:= e^{ax} [ - (b/a² + b²) cos(bx + c) + (a/(a² + b²)) sin(bx + c) ].Wait, let me check that again.Wait, the numerator after multiplying by b² is:- (1/b) * b² cos(bx + c) + (a/b²) * b² sin(bx + c) = -b cos(bx + c) + a sin(bx + c).So, the integral is e^{ax} [ -b cos(bx + c) + a sin(bx + c) ] / (a² + b²).Yes, that's correct.Therefore, ∫ e^{ax} sin(bx + c) dx = e^{ax} [ a sin(bx + c) - b cos(bx + c) ] / (a² + b²) + C.So, in our case, a = -k, b = ω, c = φ.Therefore, ∫ e^{-kx} sin(ωx + φ) dx = e^{-kx} [ (-k) sin(ωx + φ) - ω cos(ωx + φ) ] / (k² + ω²) + C.So, that's the indefinite integral. Now, we need to evaluate it from 0 to t.So, the definite integral from 0 to t is:[ e^{-k t} ( -k sin(ω t + φ) - ω cos(ω t + φ) ) / (k² + ω²) ] - [ e^{0} ( -k sin(φ) - ω cos(φ) ) / (k² + ω²) ].Simplify this:= [ e^{-k t} ( -k sin(ω t + φ) - ω cos(ω t + φ) ) - ( -k sin(φ) - ω cos(φ) ) ] / (k² + ω²).Factor out the negative sign:= [ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] / (k² + ω²).So, putting it all together, the first integral is:A times that expression, so:A [ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] / (k² + ω²).Therefore, R(t) is this expression plus Bt.So, R(t) = [ A ( - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ) ] / (k² + ω²) + Bt.That's the general form of R(t). Let me write it neatly:R(t) = (A / (k² + ω²)) [ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] + Bt.I think that's the integral. Let me double-check the signs.When I did the integral, I had:∫ e^{-kx} sin(ωx + φ) dx = e^{-kx} [ -k sin(ωx + φ) - ω cos(ωx + φ) ] / (k² + ω²) + C.So, when evaluating from 0 to t, it's:[ e^{-k t} ( -k sin(ω t + φ) - ω cos(ω t + φ) ) / (k² + ω²) ] - [ e^{0} ( -k sin(φ) - ω cos(φ) ) / (k² + ω²) ].Which is:[ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) / (k² + ω²) ] + [ (k sin(φ) + ω cos(φ)) / (k² + ω²) ].So, factoring out 1/(k² + ω²), we get:[ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] / (k² + ω²).Then, multiplying by A, we have:A [ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] / (k² + ω²).Adding the integral of B, which is Bt, gives the final expression for R(t).So, I think that's correct. Let me just write it again:R(t) = (A / (k² + ω²)) [ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] + Bt.I think that's the general form.So, summarizing:1. From the maximum discipline condition, we found that 50 = A e^{-kπ/(2ω)} cos(φ).2. The integral R(t) is given by the expression above.I think that's all. Let me just make sure I didn't make any sign errors.Wait, in the integral, when I did the integration by parts, I had:∫ e^{-kx} sin(ωx + φ) dx = e^{-kx} [ -k sin(ωx + φ) - ω cos(ωx + φ) ] / (k² + ω²) + C.Wait, no, actually, let me re-examine the integral formula.Earlier, I had:∫ e^{ax} sin(bx + c) dx = e^{ax} [ a sin(bx + c) - b cos(bx + c) ] / (a² + b²) + C.But in our case, a = -k, so plugging in:= e^{-kx} [ (-k) sin(ωx + φ) - ω cos(ωx + φ) ] / (k² + ω²) + C.Yes, that's correct. So, when evaluating from 0 to t, it's:[ e^{-k t} ( -k sin(ω t + φ) - ω cos(ω t + φ) ) / (k² + ω²) ] - [ e^{0} ( -k sin(φ) - ω cos(φ) ) / (k² + ω²) ].Which simplifies to:[ - e^{-k t} (k sin(ω t + φ) + ω cos(ω t + φ)) + k sin(φ) + ω cos(φ) ] / (k² + ω²).So, that's correct.Therefore, R(t) is as I wrote above.I think I'm confident with this result.Final Answer1. The relationship is boxed{50 = A e^{-frac{kpi}{2omega}} cos(phi)}.2. The general form of ( R(t) ) is boxed{R(t) = frac{A}{k^2 + omega^2} left[ -e^{-kt} (k sin(omega t + phi) + omega cos(omega t + phi)) + k sin(phi) + omega cos(phi) right] + Bt}.</think>

🧠:<think>Okay, so I have this problem where a student is trying to understand the relationship between two continuous variables, X and Y. They're using a linear regression model because they assume a bivariate normal distribution. The true relationship is given by Y = β₀ + β₁X + ε, where ε is normally distributed with mean 0 and variance σ². The student collected a sample of size n = 100. The sample means are X̄ = 50 and Ȳ = 100. The sample variances are s_XX = 2500 and s_YY = 5000, and the covariance s_XY = 3000. There are two sub-problems: first, to calculate the least squares estimates of β₀ and β₁, and second, to test the hypothesis that β₁ = 0 against the alternative β₁ ≠ 0 at a 0.05 significance level.Starting with the first sub-problem: calculating the least squares estimates. I remember that in linear regression, the slope coefficient β₁ is estimated using the covariance of X and Y divided by the variance of X. The intercept β₀ is then estimated as Ȳ - β₁*X̄.So, let's write that down. The formula for the slope estimate is:hat{β}_1 = s_{XY} / s_{XX}Given that s_{XY} is 3000 and s_{XX} is 2500, plugging those in:hat{β}_1 = 3000 / 2500 = 1.2Wait, that seems straightforward. So, the slope is 1.2. Now, for the intercept, hat{β}_0 = Ȳ - hat{β}_1 * X̄. Plugging in the numbers: Ȳ is 100, X̄ is 50, and hat{β}_1 is 1.2.So, hat{β}_0 = 100 - 1.2 * 50 = 100 - 60 = 40.So, the least squares estimates are hat{β}_0 = 40 and hat{β}_1 = 1.2.Wait, let me double-check that. The formula for the slope is covariance over variance of X, which is 3000/2500. 3000 divided by 2500 is indeed 1.2. Then, the intercept is the mean of Y minus the slope times the mean of X, so 100 - 1.2*50. 1.2*50 is 60, so 100 - 60 is 40. Yep, that seems correct.So, that's the first part done. Now, moving on to the second sub-problem: testing whether β₁ is significantly different from zero.To test the hypothesis H₀: β₁ = 0 vs. H₁: β₁ ≠ 0, we need to perform a t-test. The test statistic is calculated as:t = hat{β}_1 / SE(hat{β}_1)Where SE(hat{β}_1) is the standard error of the slope estimate.First, I need to calculate the standard error of hat{β}_1. The formula for SE(hat{β}_1) is sqrt(MSE / s_{XX}), where MSE is the mean squared error.Wait, but do I have MSE? I don't think it's provided directly. Alternatively, I can compute MSE using the residuals. But since we don't have the actual data points, maybe I can compute it using the given sums.Alternatively, I remember that in simple linear regression, the variance of the slope estimator can also be expressed as:Var(hat{β}_1) = σ² / s_{XX}But since σ² is unknown, we estimate it with MSE, which is the residual mean square. But how do I compute MSE? Let me recall that in simple linear regression, the total sum of squares (SST) is equal to the regression sum of squares (SSR) plus the residual sum of squares (SSE). SST = SSR + SSEWe can compute SST as s_YY, which is 5000. SSR is the sum of squares explained by the regression, which is hat{β}_1² * s_{XX}. So, SSR = (1.2)² * 2500 = 1.44 * 2500 = 3600.Therefore, SSE = SST - SSR = 5000 - 3600 = 1400.Then, MSE is SSE divided by the degrees of freedom, which is n - 2 (since we estimated two parameters, β₀ and β₁). So, MSE = 1400 / (100 - 2) = 1400 / 98 ≈ 14.2857.Therefore, the standard error of hat{β}_1 is sqrt(MSE / s_{XX}) = sqrt(14.2857 / 2500).Calculating that: 14.2857 divided by 2500 is approximately 0.00571428. The square root of that is approximately 0.0756.So, SE(hat{β}_1) ≈ 0.0756.Now, the t-statistic is hat{β}_1 / SE(hat{β}_1) = 1.2 / 0.0756 ≈ 15.86.Wait, that seems quite large. Let me verify my calculations.First, SSR = (1.2)^2 * 2500 = 1.44 * 2500 = 3600. Correct.SSE = 5000 - 3600 = 1400. Correct.MSE = 1400 / 98 ≈ 14.2857. Correct.Then, SE(hat{β}_1) = sqrt(MSE / s_{XX}) = sqrt(14.2857 / 2500). Let me compute 14.2857 / 2500:14.2857 divided by 2500 is 0.00571428. The square root of that is approximately 0.0756. Correct.So, t = 1.2 / 0.0756 ≈ 15.86. That is indeed a very large t-statistic.Now, the degrees of freedom for the t-test is n - 2 = 98. Looking up the critical t-value for a two-tailed test at α = 0.05 with 98 degrees of freedom. Since 98 is a large number, the critical value is approximately 1.984 (using the t-table or z-table approximation, since for large df, t approaches z). But our calculated t-statistic is 15.86, which is way larger than 1.984. Therefore, we can reject the null hypothesis at the 0.05 significance level. Alternatively, we can compute the p-value. Given such a large t-statistic, the p-value will be extremely small, much less than 0.05. So, we have strong evidence to reject H₀ and conclude that β₁ is significantly different from zero.Wait, but let me think again. Did I compute the standard error correctly? Because sometimes, the formula for SE(hat{β}_1) is sqrt(MSE / (s_{XX}/(n-1))). Wait, no, actually, s_{XX} is already the sum of squared deviations, which is (n-1) times the sample variance. So, if s_{XX} = 2500, then the sample variance of X is s_{XX}/(n-1) = 2500/99 ≈ 25.25. But in the formula for Var(hat{β}_1), it's σ² / s_{XX}. Since σ² is estimated by MSE, which is already an average over n-2, so MSE is in units of variance. Therefore, SE(hat{β}_1) is sqrt(MSE / s_{XX}).Wait, but let me double-check the formula. The variance of the slope estimator is Var(hat{β}_1) = σ² / Σ(X_i - X̄)^2. Since Σ(X_i - X̄)^2 is s_{XX}, which is 2500. So, Var(hat{β}_1) = σ² / 2500. But σ² is estimated by MSE, which is 14.2857. Therefore, Var(hat{β}_1) = 14.2857 / 2500 ≈ 0.00571428. Therefore, SE(hat{β}_1) is sqrt(0.00571428) ≈ 0.0756. So, that part is correct.Therefore, the t-statistic is indeed 1.2 / 0.0756 ≈ 15.86. That's a huge t-value, so the p-value is practically zero. Hence, we reject the null hypothesis.Alternatively, another way to compute the standard error is using the formula:SE(hat{β}_1) = sqrt[(1/(n-2)) * (s_YY - hat{β}_1² * s_{XX}) / s_{XX}]Wait, let me see. Let me compute that:SE(hat{β}_1) = sqrt[(1/(n-2)) * (s_YY - hat{β}_1² * s_{XX}) / s_{XX}]Plugging in the numbers:s_YY = 5000, hat{β}_1² = 1.44, s_{XX}=2500, n-2=98.So, numerator inside the sqrt is (1/98) * (5000 - 1.44*2500) / 2500.Compute 1.44*2500 = 3600.So, 5000 - 3600 = 1400.Then, 1400 / 2500 = 0.56.Then, (1/98)*0.56 ≈ 0.00571428.So, sqrt(0.00571428) ≈ 0.0756. Same result. So, that's consistent.Therefore, the t-statistic is 1.2 / 0.0756 ≈ 15.86.Given that, the p-value is extremely small, so we reject H₀.Alternatively, another way to think about it is using the correlation coefficient. The correlation r is s_XY / sqrt(s_XX * s_YY) = 3000 / sqrt(2500*5000) = 3000 / sqrt(12,500,000) = 3000 / 3535.53 ≈ 0.8485.Then, the t-statistic can also be calculated as r * sqrt(n - 2) / sqrt(1 - r²).Let me compute that:r ≈ 0.8485n - 2 = 981 - r² ≈ 1 - 0.7198 ≈ 0.2802So, t = 0.8485 * sqrt(98) / sqrt(0.2802)Compute sqrt(98) ≈ 9.8995sqrt(0.2802) ≈ 0.5293So, t ≈ 0.8485 * 9.8995 / 0.5293 ≈ (8.399) / 0.5293 ≈ 15.86. Same result.So, that confirms the t-statistic is indeed 15.86, which is way beyond the critical value, so we reject the null hypothesis.Therefore, the regression coefficient hat{β}_1 is statistically significant at the 0.05 level.Wait, just to make sure I didn't make any calculation errors. Let me recompute the t-statistic using the correlation method.r = 3000 / sqrt(2500*5000) = 3000 / sqrt(12,500,000) = 3000 / 3535.53 ≈ 0.8485. Correct.Then, t = r * sqrt(n - 2) / sqrt(1 - r²)Compute 1 - r²: 1 - (0.8485)^2 ≈ 1 - 0.7198 ≈ 0.2802. Correct.sqrt(1 - r²) ≈ sqrt(0.2802) ≈ 0.5293. Correct.sqrt(n - 2) = sqrt(98) ≈ 9.8995. Correct.So, t ≈ 0.8485 * 9.8995 / 0.5293 ≈ (8.399) / 0.5293 ≈ 15.86. Correct.So, all methods lead to the same t-statistic, which is very large, so the p-value is practically zero.Therefore, the conclusion is that β₁ is significantly different from zero.I think that's solid. So, summarizing:1. The least squares estimates are hat{β}_0 = 40 and hat{β}_1 = 1.2.2. The t-statistic is approximately 15.86, which is much larger than the critical value of approximately 1.984 at α = 0.05 with 98 degrees of freedom. Therefore, we reject the null hypothesis and conclude that β₁ is statistically significant.</think>