🧠:<think>Okay, so I have two problems to solve here. Let me take them one at a time.Starting with problem 1: Anita wants to maximize the total voter turnout, which is the dot product of the registered voters vector v and the turnout rates vector t. The constraint is that the sum of the squares of the turnout rates is exactly 2. So, mathematically, we're trying to maximize v ⋅ t subject to the condition that ||t||² = 2.Hmm, this sounds like an optimization problem with a constraint. I remember that in such cases, we can use the method of Lagrange multipliers. The idea is to find the maximum of the function f(t) = v ⋅ t under the constraint g(t) = t₁² + t₂² + t₃² + t₄² + t₅² - 2 = 0.So, setting up the Lagrangian, L = v ⋅ t - λ(g(t)). That would be L = v₁t₁ + v₂t₂ + v₃t₃ + v₄t₄ + v₅t₅ - λ(t₁² + t₂² + t₃² + t₄² + t₅² - 2).To find the maximum, we take the partial derivatives of L with respect to each tᵢ and set them equal to zero. So, for each i from 1 to 5, ∂L/∂tᵢ = vᵢ - 2λtᵢ = 0. Solving for tᵢ, we get tᵢ = vᵢ / (2λ).Now, since all tᵢ are expressed in terms of λ, we can substitute back into the constraint equation to solve for λ. The constraint is t₁² + t₂² + t₃² + t₄² + t₅² = 2. Substituting tᵢ = vᵢ / (2λ) gives:(v₁² + v₂² + v₃² + v₄² + v₅²) / (4λ²) = 2.Let me denote the sum of squares of v as S = v₁² + v₂² + v₃² + v₄² + v₅². Then, S / (4λ²) = 2, so 4λ² = S / 2, which means λ² = S / 8, so λ = sqrt(S / 8). Since λ is a positive constant (because it's a multiplier for the constraint which is positive), we can write λ = sqrt(S) / (2√2).Therefore, substituting back into tᵢ, we get tᵢ = vᵢ / (2 * (sqrt(S) / (2√2))) ) = vᵢ * (2√2) / (2 sqrt(S)) ) = vᵢ * √2 / sqrt(S).Simplifying, tᵢ = (vᵢ / sqrt(S)) * √2. So, each tᵢ is proportional to vᵢ, scaled by √2 divided by the norm of v.Wait, let me double-check that. So, tᵢ = (vᵢ / (2λ)) and λ = sqrt(S) / (2√2). So, 2λ = sqrt(S) / √2. Therefore, tᵢ = vᵢ / (sqrt(S) / √2) ) = vᵢ * √2 / sqrt(S). Yes, that seems correct.So, the optimal turnout rates are each tᵢ = (vᵢ * √2) / ||v||, where ||v|| is the Euclidean norm of v.Let me think if there's another way to approach this. Maybe using the Cauchy-Schwarz inequality? Because the dot product is maximized when t is in the same direction as v, scaled appropriately.Indeed, the Cauchy-Schwarz inequality states that v ⋅ t ≤ ||v|| ||t||. Since we have a constraint on ||t||² = 2, so ||t|| = sqrt(2). Therefore, the maximum dot product is ||v|| * sqrt(2). This occurs when t is in the direction of v, scaled by sqrt(2) / ||v||. So, tᵢ = (vᵢ / ||v||) * sqrt(2). Which is the same result as before. So, that confirms it.Therefore, the optimal turnout rates are each tᵢ = (vᵢ * sqrt(2)) / ||v||.Moving on to problem 2: Rahul and Aryan are modeling the probability P(x) that a voter will vote as a function of campaign intensity x. The model is given by P(x) = 1 / (1 + e^{-kx}), where k > 0 is a constant. They observe that an increase of 10 units in x results in an increase of 0.1 in P(x). We need to find k and verify if this increase is consistent with the model.So, first, let's understand the model. P(x) is a logistic function, which is an S-shaped curve that increases from 0 to 1 as x increases. The parameter k controls the steepness of the curve. A higher k means a steeper curve, so the probability increases more rapidly with x.Given that increasing x by 10 units increases P(x) by 0.1, we can model this as P(x + 10) - P(x) = 0.1.So, let's write that equation:1 / (1 + e^{-k(x + 10)}) - 1 / (1 + e^{-kx}) = 0.1.Let me denote e^{-kx} as y to simplify the equation. Then, e^{-k(x + 10)} = e^{-kx} * e^{-10k} = y * e^{-10k}.So, substituting into the equation:1 / (1 + y * e^{-10k}) - 1 / (1 + y) = 0.1.Let me compute this difference:[ (1 + y) - (1 + y * e^{-10k}) ] / [ (1 + y)(1 + y * e^{-10k}) ] = 0.1.Simplify the numerator:(1 + y - 1 - y * e^{-10k}) = y (1 - e^{-10k}).So, the equation becomes:y (1 - e^{-10k}) / [ (1 + y)(1 + y * e^{-10k}) ] = 0.1.Hmm, this seems a bit complicated. Maybe instead of substituting, I can take derivatives? Because the change in P with respect to x is dP/dx = k * P(x) * (1 - P(x)). So, the derivative is proportional to k.But the problem states that an increase of 10 units in x results in an increase of 0.1 in P. So, perhaps we can approximate this change using the derivative.If Δx = 10, then ΔP ≈ dP/dx * Δx = k * P(x) * (1 - P(x)) * 10.Given that ΔP = 0.1, so 0.1 ≈ 10k P(x)(1 - P(x)).But this is an approximation, assuming that P(x) doesn't change too much over Δx. However, without knowing the value of x, we can't directly compute P(x). Alternatively, maybe we can consider the maximum rate of change.Wait, the maximum rate of change of P(x) occurs where the derivative is maximized. The derivative dP/dx = k P(x)(1 - P(x)) is maximized when P(x) = 0.5, since that's where the logistic curve has the steepest slope. At P(x) = 0.5, dP/dx = k * 0.5 * 0.5 = k/4.So, the maximum rate of change is k/4. If we assume that the change occurs around this point, then ΔP ≈ (k/4) * Δx. So, 0.1 ≈ (k/4) * 10, which gives 0.1 ≈ (10k)/4, so 0.1 ≈ 2.5k, so k ≈ 0.04.But wait, this is an approximation. Let me see if this is consistent.Alternatively, maybe we can set up the equation exactly. Let me denote x as some value, then P(x + 10) - P(x) = 0.1.But without knowing x, it's hard to solve exactly. Maybe we can assume that x is such that P(x) is around 0.5, where the change is most sensitive.Alternatively, perhaps we can take the derivative and set it equal to ΔP / Δx, which is 0.1 / 10 = 0.01. So, dP/dx = 0.01.But dP/dx = k P(x)(1 - P(x)) = 0.01.So, k = 0.01 / [P(x)(1 - P(x))].But again, without knowing P(x), we can't get k. Unless we assume that P(x) is 0.5, which would give k = 0.01 / (0.25) = 0.04. So, that's consistent with the previous approximation.But is this the only way? Maybe not. Alternatively, perhaps we can consider the function P(x + 10) - P(x) = 0.1 and solve for k.Let me write the equation again:1 / (1 + e^{-k(x + 10)}) - 1 / (1 + e^{-kx}) = 0.1.Let me denote z = e^{-kx}. Then, e^{-k(x + 10)} = z * e^{-10k}.So, substituting:1 / (1 + z * e^{-10k}) - 1 / (1 + z) = 0.1.Let me compute this:[ (1 + z) - (1 + z * e^{-10k}) ] / [ (1 + z)(1 + z * e^{-10k}) ] = 0.1.Simplify numerator:1 + z - 1 - z * e^{-10k} = z (1 - e^{-10k}).So, the equation becomes:z (1 - e^{-10k}) / [ (1 + z)(1 + z * e^{-10k}) ] = 0.1.Hmm, this still has z in it, which is e^{-kx}. Without knowing x, we can't solve for z. So, perhaps we need another approach.Wait, maybe we can consider the derivative approach. The average rate of change over the interval x to x + 10 is 0.1 / 10 = 0.01. So, the average derivative over that interval is 0.01. So, integrating dP/dx over x to x + 10 gives 0.1.But dP/dx = k P (1 - P). So, ∫_{x}^{x + 10} k P(t)(1 - P(t)) dt = 0.1.This integral is equal to 0.1. But solving this integral exactly is complicated because it's a function of P(t), which itself depends on t.Alternatively, maybe we can approximate the integral using the average value of P(t) over the interval. If we assume that P(t) is roughly constant over the interval, say P_avg, then the integral becomes k P_avg (1 - P_avg) * 10 = 0.1. So, k P_avg (1 - P_avg) = 0.01.But again, without knowing P_avg, we can't solve for k. However, if we assume that P_avg is around 0.5, which is where the derivative is maximized, then k * 0.25 = 0.01, so k = 0.04. This is the same as before.Alternatively, maybe we can solve the equation numerically. Let's assume that P(x) is 0.5 at some point x, then P(x + 10) would be 0.6, since it increased by 0.1. Let's check if that's consistent.If P(x) = 0.5, then 0.5 = 1 / (1 + e^{-kx}), so e^{-kx} = 1, so kx = 0, which implies x = 0. Wait, that can't be right because if x = 0, then P(0) = 0.5, and P(10) = 1 / (1 + e^{-10k}).We want P(10) = 0.6, so 0.6 = 1 / (1 + e^{-10k}), so 1 + e^{-10k} = 1 / 0.6 ≈ 1.6667, so e^{-10k} ≈ 0.6667, so -10k ≈ ln(0.6667) ≈ -0.4055, so k ≈ 0.4055 / 10 ≈ 0.04055.So, k ≈ 0.04055. That's approximately 0.0406, which is close to 0.04. So, this seems consistent.Therefore, k is approximately 0.0406. To verify, let's compute P(10) with k = 0.0406:P(10) = 1 / (1 + e^{-0.0406 * 10}) = 1 / (1 + e^{-0.406}) ≈ 1 / (1 + 0.6667) ≈ 1 / 1.6667 ≈ 0.6. So, that works.Therefore, k ≈ 0.0406. To be precise, let's solve for k exactly.We have P(x + 10) - P(x) = 0.1.Assuming that P(x) = 0.5, then P(x + 10) = 0.6.So, 0.6 = 1 / (1 + e^{-k(x + 10)}).But since P(x) = 0.5, we have 0.5 = 1 / (1 + e^{-kx}), so e^{-kx} = 1, so kx = 0, so x = 0.Therefore, P(10) = 0.6 = 1 / (1 + e^{-10k}).So, 1 + e^{-10k} = 1 / 0.6 ≈ 1.6667.Thus, e^{-10k} ≈ 0.6667.Taking natural log: -10k ≈ ln(0.6667) ≈ -0.4055.Therefore, k ≈ 0.4055 / 10 ≈ 0.04055.So, k ≈ 0.04055, which is approximately 0.0406.Therefore, the value of k is approximately 0.0406, and the increase is consistent with the model because when we plug k back into the equation, we get the desired increase in P(x) when x increases by 10 units.</think>

🧠:<think>Okay, so I've got this problem about a linguist analyzing verb tenses in a corpus. It's divided into two parts. Let me tackle them one by one.Starting with part 1: The linguist is using a multinomial distribution to model the occurrence of past, present, and future tenses. The corpus has 50,000 sentences, with 20,000 past, 25,000 present, and 5,000 future. I need to find the likelihood function and then the MLEs for the parameters θ₁, θ₂, θ₃.Hmm, multinomial distribution. The likelihood function for multinomial data is given by the product of the probabilities for each outcome raised to the number of times they occur. So, the formula is:L(θ₁, θ₂, θ₃) = (N! / (n₁! n₂! n₃!)) * (θ₁^n₁ θ₂^n₂ θ₃^n₃)Where N is the total number of sentences, which is 50,000, and n₁, n₂, n₃ are the counts for past, present, future respectively. Plugging in the numbers:L = (50,000! / (20,000! 25,000! 5,000!)) * (θ₁^20,000 θ₂^25,000 θ₃^5,000)But since the multinomial coefficients are constants with respect to θ, the MLEs are found by maximizing the product θ₁^n₁ θ₂^n₂ θ₃^n₃, subject to θ₁ + θ₂ + θ₃ = 1.I remember that for multinomial MLEs, each parameter is just the count divided by the total. So θ₁ = n₁/N, θ₂ = n₂/N, θ₃ = n₃/N.Calculating these:θ₁ = 20,000 / 50,000 = 0.4θ₂ = 25,000 / 50,000 = 0.5θ₃ = 5,000 / 50,000 = 0.1That seems straightforward.Moving on to part 2: Now, it's about a hidden Markov model (HMM). Each state is a verb tense: past, present, future. The transition probabilities are given in matrix A, and the initial distribution π is given. We need to calculate the probability of observing the sequence: past, present, future, past, present, given the initial state is past.Wait, so the initial state is past, and we have a sequence of observations: past, present, future, past, present. So the states are the hidden states, and the observations are the verb tenses. But in HMMs, usually, the observations are emitted from the states, but here, it seems like each sentence's tense is the state itself. Hmm, maybe it's a bit different.Wait, actually, in this case, each sentence's tense is the state, so the model is a bit simpler. So the sequence of tenses is the sequence of states, and we need to compute the probability of that state sequence given the initial state and the transition matrix.So, the initial state is past (T₁ = past). Then, the sequence is past, present, future, past, present. That's 5 transitions: from past to present, present to future, future to past, past to present.Wait, but the sequence has 5 tenses, so it's 4 transitions. Let me count: T₁ = past, T₂ = present, T₃ = future, T₄ = past, T₅ = present. So transitions are T₁→T₂, T₂→T₃, T₃→T₄, T₄→T₅. So four transitions.Given the transition matrix A, where A[i][j] is the probability of going from state i to state j.First, let's map the states to indices. Let's say past is state 1, present is state 2, future is state 3. So π = [0.4, 0.5, 0.1], but the initial state is past, so π is actually [1, 0, 0] because we're starting at past.Wait, no. Wait, the initial distribution π is given as (0.4, 0.5, 0.1), but the initial state is past. So actually, the initial state is fixed as past, so the initial probability is 1 for past, and 0 for others. So maybe we don't use π here because the initial state is given.So the probability is the product of the transitions:P = P(T₁ = past) * P(T₂ = present | T₁ = past) * P(T₃ = future | T₂ = present) * P(T₄ = past | T₃ = future) * P(T₅ = present | T₄ = past)But since T₁ is given as past, P(T₁ = past) is 1, so the total probability is the product of the transitions:P = a_{12} * a_{23} * a_{31} * a_{12}Wait, let's confirm:From past (1) to present (2): a_{12} = 0.3From present (2) to future (3): a_{23} = 0.3From future (3) to past (1): a_{31} = 0.1From past (1) to present (2): a_{12} = 0.3So multiplying these together:P = 0.3 * 0.3 * 0.1 * 0.3Calculating that:0.3 * 0.3 = 0.090.09 * 0.1 = 0.0090.009 * 0.3 = 0.0027So the probability is 0.0027.But wait, let me double-check the transition indices. The transition matrix A is:Row 1: a11=0.6, a12=0.3, a13=0.1Row 2: a21=0.2, a22=0.5, a23=0.3Row 3: a31=0.1, a32=0.4, a33=0.5So from past (1) to present (2): a12=0.3From present (2) to future (3): a23=0.3From future (3) to past (1): a31=0.1From past (1) to present (2): a12=0.3Yes, that's correct. So the product is 0.3 * 0.3 * 0.1 * 0.3 = 0.0027.So the probability is 0.0027.Wait, but in HMMs, sometimes the initial state is considered as part of the model, but here it's given that the initial state is past, so we don't need to multiply by the initial distribution π. We just start at past with probability 1.So yes, the calculation seems right.</think>

🧠:<think>Okay, so I have this problem about finding the coordinates of a temple T such that the centroid of triangle ATW lies on the line y = 2x + 1. Then, I also need to verify that the distance from T to A is twice the distance from T to W. Hmm, let me break this down step by step.First, let's recall what a centroid is. The centroid of a triangle is the average of the coordinates of its three vertices. So, if I have points A(3,7), W(-2,-1), and T(x,y), the centroid G would be:G = [(3 + (-2) + x)/3, (7 + (-1) + y)/3]Simplifying that, the x-coordinate of G is (1 + x)/3 and the y-coordinate is (6 + y)/3. Now, the centroid lies on the line y = 2x + 1. So, substituting the coordinates of G into this equation should satisfy it. Let me write that out:(6 + y)/3 = 2*( (1 + x)/3 ) + 1Hmm, let's solve this equation for x and y. First, multiply both sides by 3 to eliminate the denominators:6 + y = 2*(1 + x) + 3Simplify the right side:6 + y = 2 + 2x + 3Combine like terms:6 + y = 5 + 2xNow, subtract 5 from both sides:1 + y = 2xSo, y = 2x - 1. Wait, that's interesting. So, the coordinates of T must lie on the line y = 2x - 1. But that's just one equation, and we have two variables, x and y. So, I need another condition to find the exact coordinates of T.The second part of the problem says that the distance from T to A is twice the distance from T to W. So, mathematically, that is:Distance(T, A) = 2 * Distance(T, W)Let me write the distance formulas:Distance(T, A) = sqrt[(x - 3)^2 + (y - 7)^2]Distance(T, W) = sqrt[(x + 2)^2 + (y + 1)^2]So, setting up the equation:sqrt[(x - 3)^2 + (y - 7)^2] = 2 * sqrt[(x + 2)^2 + (y + 1)^2]To eliminate the square roots, I'll square both sides:(x - 3)^2 + (y - 7)^2 = 4 * [(x + 2)^2 + (y + 1)^2]Let me expand both sides:Left side:(x^2 - 6x + 9) + (y^2 - 14y + 49) = x^2 - 6x + 9 + y^2 - 14y + 49 = x^2 + y^2 - 6x - 14y + 58Right side:4 * [(x^2 + 4x + 4) + (y^2 + 2y + 1)] = 4*(x^2 + 4x + 4 + y^2 + 2y + 1) = 4x^2 + 16x + 16 + 4y^2 + 8y + 4 = 4x^2 + 4y^2 + 16x + 8y + 20So, the equation becomes:x^2 + y^2 - 6x - 14y + 58 = 4x^2 + 4y^2 + 16x + 8y + 20Let me bring all terms to the left side:x^2 + y^2 - 6x - 14y + 58 - 4x^2 - 4y^2 - 16x - 8y - 20 = 0Combine like terms:(1 - 4)x^2 + (1 - 4)y^2 + (-6 - 16)x + (-14 - 8)y + (58 - 20) = 0Which simplifies to:-3x^2 - 3y^2 - 22x - 22y + 38 = 0Hmm, that's a bit messy. Maybe I can divide the entire equation by -1 to make the coefficients positive:3x^2 + 3y^2 + 22x + 22y - 38 = 0Hmm, perhaps factor out a 3 from the x^2 and y^2 terms:3(x^2 + y^2) + 22x + 22y - 38 = 0Not sure if that helps. Maybe I can use the equation we got from the centroid condition, which was y = 2x - 1, and substitute that into this equation.So, substituting y = 2x - 1 into 3x^2 + 3y^2 + 22x + 22y - 38 = 0:First, compute y^2:y = 2x - 1, so y^2 = (2x - 1)^2 = 4x^2 - 4x + 1Now, substitute:3x^2 + 3*(4x^2 - 4x + 1) + 22x + 22*(2x - 1) - 38 = 0Let me expand each term:3x^2 + 12x^2 - 12x + 3 + 22x + 44x - 22 - 38 = 0Combine like terms:(3x^2 + 12x^2) + (-12x + 22x + 44x) + (3 - 22 - 38) = 0Calculating each:15x^2 + (54x) + (-57) = 0So, the equation is:15x^2 + 54x - 57 = 0Let me see if I can simplify this equation. All coefficients are divisible by 3:Divide by 3:5x^2 + 18x - 19 = 0Now, this is a quadratic equation in x. Let's use the quadratic formula to solve for x.Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a)Here, a = 5, b = 18, c = -19Compute discriminant:b^2 - 4ac = (18)^2 - 4*5*(-19) = 324 + 380 = 704So, sqrt(704). Let me simplify sqrt(704):704 = 64 * 11, so sqrt(704) = 8*sqrt(11)So, x = [ -18 ± 8*sqrt(11) ] / (2*5) = [ -18 ± 8*sqrt(11) ] / 10Simplify numerator:Factor out 2: 2*(-9 ± 4*sqrt(11)) / 10 = (-9 ± 4*sqrt(11))/5So, x = (-9 + 4*sqrt(11))/5 or x = (-9 - 4*sqrt(11))/5Now, let's find the corresponding y values using y = 2x - 1.First, for x = (-9 + 4*sqrt(11))/5:y = 2*( (-9 + 4*sqrt(11))/5 ) - 1 = (-18 + 8*sqrt(11))/5 - 5/5 = (-23 + 8*sqrt(11))/5Second, for x = (-9 - 4*sqrt(11))/5:y = 2*( (-9 - 4*sqrt(11))/5 ) - 1 = (-18 - 8*sqrt(11))/5 - 5/5 = (-23 - 8*sqrt(11))/5So, the possible coordinates for T are:T1 = ( (-9 + 4*sqrt(11))/5 , (-23 + 8*sqrt(11))/5 )andT2 = ( (-9 - 4*sqrt(11))/5 , (-23 - 8*sqrt(11))/5 )Now, I need to verify which of these points actually satisfy the distance condition. Wait, actually, since we derived these points based on both the centroid condition and the distance condition, both should satisfy. But let me check.Wait, actually, when we squared both sides of the distance equation, we might have introduced extraneous solutions. So, it's a good idea to check both points.But before that, let me compute the numerical values to get an idea.Compute sqrt(11) ≈ 3.3166Compute for T1:x = (-9 + 4*3.3166)/5 ≈ (-9 + 13.2664)/5 ≈ 4.2664/5 ≈ 0.8533y = (-23 + 8*3.3166)/5 ≈ (-23 + 26.5328)/5 ≈ 3.5328/5 ≈ 0.7066So, T1 ≈ (0.8533, 0.7066)Compute for T2:x = (-9 - 4*3.3166)/5 ≈ (-9 - 13.2664)/5 ≈ (-22.2664)/5 ≈ -4.4533y = (-23 - 8*3.3166)/5 ≈ (-23 - 26.5328)/5 ≈ (-49.5328)/5 ≈ -9.9066So, T2 ≈ (-4.4533, -9.9066)Now, let's check the distance condition for T1.Compute distance from T1 to A(3,7):sqrt[(0.8533 - 3)^2 + (0.7066 - 7)^2] ≈ sqrt[(-2.1467)^2 + (-6.2934)^2] ≈ sqrt[4.608 + 39.599] ≈ sqrt[44.207] ≈ 6.65Distance from T1 to W(-2,-1):sqrt[(0.8533 + 2)^2 + (0.7066 + 1)^2] ≈ sqrt[(2.8533)^2 + (1.7066)^2] ≈ sqrt[8.142 + 2.913] ≈ sqrt[11.055] ≈ 3.326Now, 6.65 is approximately twice 3.326, so that works.Now, check T2.Distance from T2 to A(3,7):sqrt[(-4.4533 - 3)^2 + (-9.9066 - 7)^2] ≈ sqrt[(-7.4533)^2 + (-16.9066)^2] ≈ sqrt[55.55 + 285.84] ≈ sqrt[341.39] ≈ 18.47Distance from T2 to W(-2,-1):sqrt[(-4.4533 + 2)^2 + (-9.9066 + 1)^2] ≈ sqrt[(-2.4533)^2 + (-8.9066)^2] ≈ sqrt[6.018 + 79.33] ≈ sqrt[85.348] ≈ 9.24Now, 18.47 is approximately twice 9.24, so that also works.Wait, so both points satisfy the distance condition. Hmm, that's interesting. So, both T1 and T2 are valid solutions. But let me think about the centroid condition.Wait, the centroid is determined by the average of the three points, so if both T1 and T2 satisfy the centroid condition, then both are valid. So, perhaps there are two possible temples? Or maybe only one is in a certain region.But the problem doesn't specify any restrictions on the location of T, so both could be possible. However, the problem says "the temple", implying a single point. Maybe I made a mistake in the algebra.Wait, let me go back. When I set up the equation for the centroid, I had:(6 + y)/3 = 2*( (1 + x)/3 ) + 1Which simplified to y = 2x - 1. That seems correct.Then, for the distance condition, I set up the equation correctly and arrived at two solutions. So, perhaps both points are valid, but maybe only one lies on the line y = 2x - 1. Wait, no, both points were derived from that line, so both lie on it.Wait, but when I calculated T1 and T2, both lie on y = 2x - 1, as expected. So, both are valid. So, perhaps the problem allows for two possible temples? Or maybe I missed something.Wait, let me check the centroid calculation for both points.For T1: (0.8533, 0.7066)Centroid G = [(3 + (-2) + 0.8533)/3, (7 + (-1) + 0.7066)/3] ≈ [(1.8533)/3, (6.7066)/3] ≈ (0.6178, 2.2355)Check if this lies on y = 2x + 1:2*0.6178 + 1 ≈ 1.2356 + 1 ≈ 2.2356, which matches the y-coordinate. So, correct.For T2: (-4.4533, -9.9066)Centroid G = [(3 + (-2) + (-4.4533))/3, (7 + (-1) + (-9.9066))/3] ≈ [(-3.4533)/3, (-3.9066)/3] ≈ (-1.1511, -1.3022)Check if this lies on y = 2x + 1:2*(-1.1511) + 1 ≈ -2.3022 + 1 ≈ -1.3022, which matches the y-coordinate. So, correct.So, both points satisfy the centroid condition and the distance condition. Therefore, there are two possible solutions for T.But the problem says "the temple", which is singular. Maybe I need to consider if both are valid or if one is extraneous. Wait, when I squared the distance equation, sometimes extraneous solutions can appear, but in this case, both solutions satisfy the original distance condition, as I checked numerically.So, perhaps the problem allows for two possible temples. But let me check the problem statement again.It says: "the temple is known to align with two other significant points... The temple's location is also known to maintain a specific ratio of distances to the library and the well." So, it's possible that both points are valid, but perhaps only one is in a certain region or satisfies additional constraints not mentioned.Alternatively, perhaps I made a mistake in the algebra when solving the equations. Let me double-check.Starting from the distance equation:sqrt[(x - 3)^2 + (y - 7)^2] = 2*sqrt[(x + 2)^2 + (y + 1)^2]Squared both sides:(x - 3)^2 + (y - 7)^2 = 4[(x + 2)^2 + (y + 1)^2]Expanded correctly:x^2 -6x +9 + y^2 -14y +49 = 4x^2 +16x +16 +4y^2 +8y +4Simplify:x^2 + y^2 -6x -14y +58 = 4x^2 +4y^2 +16x +8y +20Bring all terms to left:-3x^2 -3y^2 -22x -22y +38 =0Divide by -1:3x^2 +3y^2 +22x +22y -38=0Then, substitute y=2x-1:3x^2 +3*(4x^2 -4x +1) +22x +22*(2x -1) -38=0Which is:3x^2 +12x^2 -12x +3 +22x +44x -22 -38=0Combine:15x^2 +54x -57=0Divide by 3:5x^2 +18x -19=0Solution:x = [-18 ± sqrt(324 + 380)]/10 = [-18 ± sqrt(704)]/10 = [-18 ± 8*sqrt(11)]/10 = (-9 ±4*sqrt(11))/5So, correct.Therefore, both points are valid solutions. So, perhaps the answer is both points.But the problem says "the temple", so maybe I need to present both solutions.Alternatively, perhaps I made a mistake in interpreting the centroid condition. Let me double-check.Centroid is ( (Ax + Wx + Tx)/3, (Ay + Wy + Ty)/3 )Given A(3,7), W(-2,-1), T(x,y):Gx = (3 -2 +x)/3 = (1 +x)/3Gy = (7 -1 +y)/3 = (6 + y)/3Given that G lies on y=2x+1, so:(6 + y)/3 = 2*(1 +x)/3 +1Multiply both sides by 3:6 + y = 2*(1 +x) +36 + y = 2 + 2x +36 + y = 5 + 2xSo, y = 2x -1Correct.So, all steps seem correct, leading to two possible points.Therefore, the coordinates of the temple T are:T = ( (-9 + 4√11)/5 , (-23 + 8√11)/5 ) and T = ( (-9 - 4√11)/5 , (-23 - 8√11)/5 )But the problem says "the temple", so maybe both are acceptable. Alternatively, perhaps only one is in a certain region, but since no region is specified, both are valid.Wait, but let me think about the distance condition again. The distance from T to A is twice the distance from T to W. So, the ratio is TA = 2 TW.In our solutions, both points satisfy this ratio. So, both are valid.Therefore, the answer is both points.But the problem says "the temple", so maybe I need to present both solutions.Alternatively, perhaps I made a mistake in the algebra, but I don't see where.Wait, let me check the substitution again.After substituting y=2x-1 into the distance equation, I arrived at 5x^2 +18x -19=0, which gave two solutions. So, both are valid.Therefore, the coordinates of the temple T are two points: T1 and T2 as above.But the problem might expect both answers, so I should present both.Wait, but in the problem statement, it's part 1 and part 2. Part 1 is to determine the coordinates, part 2 is to verify the distance condition. So, perhaps both points are valid, and both should be given.Alternatively, maybe only one is valid, but I can't see why.Wait, let me check the numerical values again.For T1 ≈ (0.85, 0.71), centroid ≈ (0.62, 2.24), which lies on y=2x+1: 2*0.62 +1=2.24, correct.For T2 ≈ (-4.45, -9.91), centroid ≈ (-1.15, -1.30), which lies on y=2x+1: 2*(-1.15)+1= -2.30 +1= -1.30, correct.Both correct.Therefore, the answer is both points.But the problem says "the temple", so maybe both are acceptable. Alternatively, perhaps the problem expects both solutions.So, in conclusion, the coordinates of T are:T = ( (-9 + 4√11)/5 , (-23 + 8√11)/5 ) and T = ( (-9 - 4√11)/5 , (-23 - 8√11)/5 )But let me write them in a cleaner form.First solution:x = (-9 + 4√11)/5, y = (-23 + 8√11)/5Second solution:x = (-9 - 4√11)/5, y = (-23 - 8√11)/5So, both points are valid.</think>

🧠:<think>Alright, so I have this problem about an online retailer's website traffic and conversion rate. It's divided into two parts. Let me tackle them one by one.Starting with part 1: I need to calculate the total website traffic over the first 12 months using the function ( T(t) = 500 + 100sinleft(frac{pi}{6}tright) + 50e^{-0.1t} ). The question mentions providing an exact expression for the integral and simplifying it as much as possible. Hmm, okay.First, I remember that to find the total traffic over a period, I need to integrate the traffic function over that time. Since it's over the first 12 months, the limits of integration will be from t = 0 to t = 12.So, the integral I need to compute is:[int_{0}^{12} T(t) , dt = int_{0}^{12} left(500 + 100sinleft(frac{pi}{6}tright) + 50e^{-0.1t}right) dt]I can split this integral into three separate integrals for easier computation:[int_{0}^{12} 500 , dt + int_{0}^{12} 100sinleft(frac{pi}{6}tright) dt + int_{0}^{12} 50e^{-0.1t} dt]Let me compute each integral one by one.1. The first integral is straightforward:[int_{0}^{12} 500 , dt = 500t Big|_{0}^{12} = 500(12) - 500(0) = 6000]2. The second integral involves the sine function. The integral of ( sin(ax) ) is ( -frac{1}{a}cos(ax) ). So, applying that here:Let me denote ( a = frac{pi}{6} ), so the integral becomes:[int_{0}^{12} 100sinleft(frac{pi}{6}tright) dt = 100 left( -frac{6}{pi} cosleft(frac{pi}{6}tright) right) Big|_{0}^{12}]Simplifying:[= -frac{600}{pi} left[ cosleft(frac{pi}{6} times 12right) - cos(0) right]]Calculating the arguments inside the cosine:- ( frac{pi}{6} times 12 = 2pi )- ( cos(2pi) = 1 )- ( cos(0) = 1 )So, plugging these in:[= -frac{600}{pi} [1 - 1] = -frac{600}{pi} times 0 = 0]Interesting, the sine integral over a full period (which 12 months is, since the period of ( sinleft(frac{pi}{6}tright) ) is ( frac{2pi}{pi/6} = 12 ) months) results in zero. That makes sense because the area above the curve cancels out the area below over a full period.3. The third integral is an exponential function. The integral of ( e^{kt} ) is ( frac{1}{k}e^{kt} ). Here, we have ( e^{-0.1t} ), so k = -0.1.Thus,[int_{0}^{12} 50e^{-0.1t} dt = 50 left( frac{1}{-0.1} e^{-0.1t} right) Big|_{0}^{12}]Simplify:[= -500 left[ e^{-0.1 times 12} - e^{0} right] = -500 left[ e^{-1.2} - 1 right]]Which can be rewritten as:[= -500e^{-1.2} + 500 = 500(1 - e^{-1.2})]So, putting all three integrals together:Total traffic = 6000 + 0 + 500(1 - e^{-1.2})Simplify:Total traffic = 6000 + 500 - 500e^{-1.2} = 6500 - 500e^{-1.2}I think that's the exact expression. Let me just double-check my calculations.First integral: 500*12=6000, correct.Second integral: The integral of sine over a full period is zero, correct.Third integral: 50 divided by -0.1 is -500, evaluated from 0 to 12, so 500(1 - e^{-1.2}), correct.So, yes, the exact expression is 6500 - 500e^{-1.2} thousand visitors. Since the question asks for the total traffic, which is in thousands, so the answer is 6500 - 500e^{-1.2} thousand visitors over 12 months.Moving on to part 2: The retailer's conversion rate is given by ( C(t) = 0.5R(t) - 20 ), where ( R(t) = 80 + 20cosleft(frac{pi}{6}tright) ). I need to find the average conversion rate over the first 12 months.First, let's write the expression for C(t):[C(t) = 0.5 times R(t) - 20 = 0.5 times left(80 + 20cosleft(frac{pi}{6}tright)right) - 20]Simplify:[= 40 + 10cosleft(frac{pi}{6}tright) - 20 = 20 + 10cosleft(frac{pi}{6}tright)]So, ( C(t) = 20 + 10cosleft(frac{pi}{6}tright) ).To find the average conversion rate over the first 12 months, I need to compute the average value of C(t) over t from 0 to 12. The average value of a function over [a, b] is given by:[text{Average} = frac{1}{b - a} int_{a}^{b} C(t) dt]Here, a = 0, b = 12, so:[text{Average } C = frac{1}{12} int_{0}^{12} left(20 + 10cosleft(frac{pi}{6}tright)right) dt]Again, I can split this integral into two parts:[frac{1}{12} left[ int_{0}^{12} 20 dt + int_{0}^{12} 10cosleft(frac{pi}{6}tright) dt right]]Compute each integral.1. First integral:[int_{0}^{12} 20 dt = 20t Big|_{0}^{12} = 20 times 12 - 20 times 0 = 240]2. Second integral:The integral of ( cos(ax) ) is ( frac{1}{a}sin(ax) ). Here, a = ( frac{pi}{6} ).Thus,[int_{0}^{12} 10cosleft(frac{pi}{6}tright) dt = 10 times frac{6}{pi} sinleft(frac{pi}{6}tright) Big|_{0}^{12}]Simplify:[= frac{60}{pi} left[ sinleft(frac{pi}{6} times 12right) - sin(0) right]]Calculating the arguments:- ( frac{pi}{6} times 12 = 2pi )- ( sin(2pi) = 0 )- ( sin(0) = 0 )So, plugging in:[= frac{60}{pi} [0 - 0] = 0]Therefore, the second integral is zero.Putting it all together:Average C = (1/12)(240 + 0) = 240 / 12 = 20.Wait, that's interesting. The average conversion rate is 20%.But let me double-check. The function C(t) is 20 + 10cos(πt/6). The average of a cosine function over a full period is zero, right? Because it's symmetric. So, the average of 20 + 10cos(...) over a full period is just 20. That makes sense.So, the average conversion rate is 20%.But let me verify with the integral.Yes, the integral of the cosine term over 0 to 12 is zero, so the average is just 20. Correct.Therefore, the average conversion rate over the first 12 months is 20%.I think that's all. Let me recap:1. Total traffic: 6500 - 500e^{-1.2} thousand visitors.2. Average conversion rate: 20%.I don't see any mistakes in my calculations. The integrals were handled correctly, and the average was computed properly.Final Answer1. The total website traffic over the first 12 months is boxed{6500 - 500e^{-1.2}} thousand visitors.2. The average conversion rate over the first 12 months is boxed{20%}.</think>