🧠:<think>Alright, so I have this problem where a TV producer is planning a new show about haunted locations. They want to maximize viewership by selecting four locations that are both mysterious and have a high probability of paranormal occurrences. The functions for mystery level ( M(t) ) and probability ( P(t) ) are given for each location, and both are functions of time. The producer can only choose four locations, but since there are only four, I guess they have to choose all of them? Wait, no, the first part is about determining the time ( t ) that maximizes the combined value ( S(t) ), which is the sum of ( M_i(t) times P_i(t) ) for each location. Then, in the second part, they want to select two locations with the closest combined scores ( C_i(t) = M_i(t) + P_i(t) ) at that chosen time ( t ).Okay, let me break this down step by step.First, for each location, I have expressions for ( M_i(t) ) and ( P_i(t) ). I need to compute ( S(t) = sum_{i=1}^{4} M_i(t) times P_i(t) ). So, that means for each location, I'll multiply ( M_i(t) ) by ( P_i(t) ), and then add all four results together. Then, I need to find the time ( t ) in the interval ([0, 2pi]) that maximizes this sum ( S(t) ).Let me write down the expressions for each location:- Location 1: - ( M_1(t) = 5 + 2sin(t) ) - ( P_1(t) = 0.5 + 0.3cos(t) ) - Location 2: - ( M_2(t) = 7 + sin(2t) ) - ( P_2(t) = 0.6 + 0.2cos(2t) ) - Location 3: - ( M_3(t) = 6 + 1.5cos(t) ) - ( P_3(t) = 0.4 + 0.4sin(t) ) - Location 4: - ( M_4(t) = 8 + cos(3t) ) - ( P_4(t) = 0.7 + 0.1sin(3t) )So, for each location, I need to compute ( M_i(t) times P_i(t) ). Let's do that one by one.Location 1:( M_1(t) times P_1(t) = (5 + 2sin t)(0.5 + 0.3cos t) )Let me expand this:= 5*0.5 + 5*0.3cos t + 2sin t*0.5 + 2sin t*0.3cos t= 2.5 + 1.5cos t + sin t + 0.6sin t cos tLocation 2:( M_2(t) times P_2(t) = (7 + sin 2t)(0.6 + 0.2cos 2t) )Expanding:= 7*0.6 + 7*0.2cos 2t + sin 2t*0.6 + sin 2t*0.2cos 2t= 4.2 + 1.4cos 2t + 0.6sin 2t + 0.2sin 2t cos 2tLocation 3:( M_3(t) times P_3(t) = (6 + 1.5cos t)(0.4 + 0.4sin t) )Expanding:= 6*0.4 + 6*0.4sin t + 1.5cos t*0.4 + 1.5cos t*0.4sin t= 2.4 + 2.4sin t + 0.6cos t + 0.6sin t cos tLocation 4:( M_4(t) times P_4(t) = (8 + cos 3t)(0.7 + 0.1sin 3t) )Expanding:= 8*0.7 + 8*0.1sin 3t + cos 3t*0.7 + cos 3t*0.1sin 3t= 5.6 + 0.8sin 3t + 0.7cos 3t + 0.1sin 3t cos 3tNow, let's sum all these up to get ( S(t) ).So, adding all the constants first:Location 1: 2.5Location 2: 4.2Location 3: 2.4Location 4: 5.6Total constants: 2.5 + 4.2 + 2.4 + 5.6 = 14.7Next, the terms with (cos t):Location 1: 1.5cos tLocation 3: 0.6cos tTotal: 1.5 + 0.6 = 2.1cos tTerms with (sin t):Location 1: sin tLocation 3: 2.4sin tTotal: 1 + 2.4 = 3.4sin tTerms with (sin 2t):Location 2: 0.6sin 2tTerms with (cos 2t):Location 2: 1.4cos 2tTerms with (sin 3t):Location 4: 0.8sin 3tTerms with (cos 3t):Location 4: 0.7cos 3tNow, the cross terms:Location 1: 0.6sin t cos tLocation 2: 0.2sin 2t cos 2tLocation 3: 0.6sin t cos tLocation 4: 0.1sin 3t cos 3tLet me handle these cross terms. I remember that ( sin x cos x = frac{1}{2}sin 2x ). So:Location 1: 0.6sin t cos t = 0.3sin 2tLocation 2: 0.2sin 2t cos 2t = 0.1sin 4tLocation 3: 0.6sin t cos t = 0.3sin 2tLocation 4: 0.1sin 3t cos 3t = 0.05sin 6tSo, combining these cross terms:From Location 1 and 3: 0.3 + 0.3 = 0.6sin 2tFrom Location 2: 0.1sin 4tFrom Location 4: 0.05sin 6tSo, putting it all together, ( S(t) ) is:14.7 + 2.1cos t + 3.4sin t + 1.4cos 2t + 0.6sin 2t + 0.6sin 2t + 0.8sin 3t + 0.7cos 3t + 0.1sin 4t + 0.05sin 6tWait, hold on. Let me check:Wait, in the cross terms, I had:From Location 1: 0.3sin 2tFrom Location 2: 0.1sin 4tFrom Location 3: 0.3sin 2tFrom Location 4: 0.05sin 6tSo, the cross terms contribute:0.3sin 2t + 0.3sin 2t = 0.6sin 2tPlus 0.1sin 4t and 0.05sin 6t.But in the earlier breakdown, I had:From Location 2: 0.6sin 2tWait, no, in the initial breakdown, the cross terms were separate. Let me clarify:Wait, no, in the initial breakdown, the cross terms were:Location 1: 0.6 sin t cos t = 0.3 sin 2tLocation 2: 0.2 sin 2t cos 2t = 0.1 sin 4tLocation 3: 0.6 sin t cos t = 0.3 sin 2tLocation 4: 0.1 sin 3t cos 3t = 0.05 sin 6tSo, the cross terms add up to:0.3 sin 2t + 0.1 sin 4t + 0.3 sin 2t + 0.05 sin 6tWhich is:(0.3 + 0.3) sin 2t + 0.1 sin 4t + 0.05 sin 6t= 0.6 sin 2t + 0.1 sin 4t + 0.05 sin 6tSo, going back to the total ( S(t) ):14.7 (constants)+ 2.1 cos t+ 3.4 sin t+ 1.4 cos 2t+ 0.6 sin 2t (from cross terms)+ 0.8 sin 3t+ 0.7 cos 3t+ 0.1 sin 4t+ 0.05 sin 6tWait, but hold on, in the initial breakdown, the cross terms were separate. So, actually, the 0.6 sin 2t is already included in the cross terms, but in the earlier breakdown, I had:From Location 2: 0.6 sin 2tWait, no, actually, in the initial breakdown of each location's product, Location 2 had 0.6 sin 2t, which is separate from the cross term 0.1 sin 4t.Similarly, Location 1 had sin t, Location 3 had 2.4 sin t, etc.So, actually, the cross terms add to 0.6 sin 2t + 0.1 sin 4t + 0.05 sin 6t.But in the earlier breakdown, the main terms from each location's product were:From Location 1: 2.5 + 1.5 cos t + sin t + 0.6 sin t cos tFrom Location 2: 4.2 + 1.4 cos 2t + 0.6 sin 2t + 0.2 sin 2t cos 2tFrom Location 3: 2.4 + 2.4 sin t + 0.6 cos t + 0.6 sin t cos tFrom Location 4: 5.6 + 0.8 sin 3t + 0.7 cos 3t + 0.1 sin 3t cos 3tSo, when we sum all the terms, we have:Constants: 2.5 + 4.2 + 2.4 + 5.6 = 14.7cos t terms: 1.5 cos t (Location 1) + 0.6 cos t (Location 3) = 2.1 cos tsin t terms: sin t (Location 1) + 2.4 sin t (Location 3) = 3.4 sin tcos 2t terms: 1.4 cos 2t (Location 2)sin 2t terms: 0.6 sin 2t (Location 2)sin 3t terms: 0.8 sin 3t (Location 4)cos 3t terms: 0.7 cos 3t (Location 4)Now, cross terms:From Location 1: 0.6 sin t cos t = 0.3 sin 2tFrom Location 2: 0.2 sin 2t cos 2t = 0.1 sin 4tFrom Location 3: 0.6 sin t cos t = 0.3 sin 2tFrom Location 4: 0.1 sin 3t cos 3t = 0.05 sin 6tSo, cross terms:0.3 sin 2t (Location 1) + 0.1 sin 4t (Location 2) + 0.3 sin 2t (Location 3) + 0.05 sin 6t (Location 4)So, combining these:sin 2t: 0.3 + 0.3 = 0.6 sin 2tsin 4t: 0.1 sin 4tsin 6t: 0.05 sin 6tTherefore, the total ( S(t) ) is:14.7 + 2.1 cos t + 3.4 sin t + 1.4 cos 2t + 0.6 sin 2t + 0.8 sin 3t + 0.7 cos 3t + 0.1 sin 4t + 0.05 sin 6tWait, but hold on, in the initial breakdown, the cross terms were 0.6 sin 2t, 0.1 sin 4t, 0.05 sin 6t, but in the main terms, we already have 0.6 sin 2t from Location 2. So, does that mean we have 0.6 sin 2t (from main) + 0.6 sin 2t (from cross terms)? That would be 1.2 sin 2t?Wait, no, let me double-check.Wait, in the main terms, from Location 2, we have 0.6 sin 2t. Then, from the cross terms, we have another 0.6 sin 2t (from Location 1 and 3). So, actually, total sin 2t terms are 0.6 + 0.6 = 1.2 sin 2t.Similarly, the cross terms add 0.1 sin 4t and 0.05 sin 6t.So, correcting that, the total ( S(t) ) is:14.7 + 2.1 cos t + 3.4 sin t + 1.4 cos 2t + 1.2 sin 2t + 0.8 sin 3t + 0.7 cos 3t + 0.1 sin 4t + 0.05 sin 6tYes, that seems correct.So, now, ( S(t) ) is a function composed of multiple sinusoidal terms with different frequencies. To find the maximum of ( S(t) ), we can consider taking its derivative and setting it to zero. However, given the complexity of the function, this might be challenging analytically. Alternatively, we can use calculus or numerical methods to find the maximum.But since this is a problem-solving scenario, perhaps we can consider simplifying or approximating.Alternatively, we can note that ( S(t) ) is a sum of sinusoids with different frequencies, so it's a periodic function, and its maximum can be found by evaluating it at critical points or using numerical optimization.But since this is a math problem, perhaps we can consider evaluating ( S(t) ) at specific points where the function might attain its maximum.Alternatively, maybe we can express ( S(t) ) in terms of multiple angles and then find its maximum.But given the multiple frequencies (1, 2, 3, 4, 6), it's quite complex. Maybe another approach is to consider that each term in ( S(t) ) can be expressed as a Fourier series, and the maximum occurs when all the sine and cosine terms are aligned to contribute constructively.But that might be too vague.Alternatively, perhaps we can compute ( S(t) ) numerically at several points in the interval [0, 2π] and find the maximum.Given that, perhaps I can compute ( S(t) ) at intervals of, say, π/6 (30 degrees) and see where it's maximized.But since I'm doing this manually, let me see if I can find critical points by taking the derivative.So, let's compute ( S'(t) ), the derivative of ( S(t) ) with respect to t.Given:( S(t) = 14.7 + 2.1 cos t + 3.4 sin t + 1.4 cos 2t + 1.2 sin 2t + 0.8 sin 3t + 0.7 cos 3t + 0.1 sin 4t + 0.05 sin 6t )So, the derivative ( S'(t) ) is:-2.1 sin t + 3.4 cos t - 2.8 sin 2t + 2.4 cos 2t + 2.4 cos 3t - 2.1 sin 3t + 0.4 cos 4t + 0.3 cos 6tWait, let me compute term by term:- The derivative of 14.7 is 0.- The derivative of 2.1 cos t is -2.1 sin t.- The derivative of 3.4 sin t is 3.4 cos t.- The derivative of 1.4 cos 2t is -2.8 sin 2t.- The derivative of 1.2 sin 2t is 2.4 cos 2t.- The derivative of 0.8 sin 3t is 2.4 cos 3t.- The derivative of 0.7 cos 3t is -2.1 sin 3t.- The derivative of 0.1 sin 4t is 0.4 cos 4t.- The derivative of 0.05 sin 6t is 0.3 cos 6t.So, putting it all together:( S'(t) = -2.1 sin t + 3.4 cos t - 2.8 sin 2t + 2.4 cos 2t + 2.4 cos 3t - 2.1 sin 3t + 0.4 cos 4t + 0.3 cos 6t )To find the critical points, we set ( S'(t) = 0 ). However, solving this equation analytically is quite complex due to the multiple frequencies. Therefore, a numerical approach is more feasible.Alternatively, perhaps we can approximate the maximum by evaluating ( S(t) ) at several points and see where it peaks.Let me create a table of ( t ) values from 0 to 2π, spaced at intervals of π/6 (30 degrees), and compute ( S(t) ) for each.But since this is time-consuming, maybe I can compute ( S(t) ) at key points where the sine and cosine functions reach their maxima or zeros.Alternatively, perhaps I can consider that the maximum occurs when all the sine and cosine terms are at their maximum, but given the different frequencies, it's unlikely they all align. However, maybe at t = π/2 or t = 0, etc.Let me compute ( S(t) ) at t = 0, π/2, π, 3π/2, 2π.At t = 0:Compute each term:- 14.7- 2.1 cos 0 = 2.1*1 = 2.1- 3.4 sin 0 = 0- 1.4 cos 0 = 1.4*1 = 1.4- 1.2 sin 0 = 0- 0.8 sin 0 = 0- 0.7 cos 0 = 0.7*1 = 0.7- 0.1 sin 0 = 0- 0.05 sin 0 = 0So, total S(0) = 14.7 + 2.1 + 1.4 + 0.7 = 14.7 + 2.1 = 16.8; 16.8 + 1.4 = 18.2; 18.2 + 0.7 = 18.9At t = π/2:Compute each term:- 14.7- 2.1 cos(π/2) = 0- 3.4 sin(π/2) = 3.4*1 = 3.4- 1.4 cos(π) = 1.4*(-1) = -1.4- 1.2 sin(π) = 0- 0.8 sin(3π/2) = 0.8*(-1) = -0.8- 0.7 cos(3π/2) = 0- 0.1 sin(2π) = 0- 0.05 sin(3π) = 0So, total S(π/2) = 14.7 + 0 + 3.4 -1.4 -0.8 + 0 + 0 + 0 = 14.7 + 3.4 = 18.1; 18.1 -1.4 = 16.7; 16.7 -0.8 = 15.9At t = π:Compute each term:- 14.7- 2.1 cos π = -2.1- 3.4 sin π = 0- 1.4 cos 2π = 1.4*1 = 1.4- 1.2 sin 2π = 0- 0.8 sin 3π = 0- 0.7 cos 3π = -0.7- 0.1 sin 4π = 0- 0.05 sin 6π = 0So, total S(π) = 14.7 -2.1 + 1.4 -0.7 = 14.7 -2.1 = 12.6; 12.6 +1.4 =14; 14 -0.7=13.3At t = 3π/2:Compute each term:- 14.7- 2.1 cos(3π/2) = 0- 3.4 sin(3π/2) = 3.4*(-1) = -3.4- 1.4 cos(3π) = 1.4*(-1) = -1.4- 1.2 sin(3π) = 0- 0.8 sin(9π/2) = 0.8*1 = 0.8 (since sin(9π/2)=sin(π/2)=1)Wait, hold on, sin(3t) at t=3π/2 is sin(9π/2) = sin(π/2) = 1Similarly, cos(3t) at t=3π/2 is cos(9π/2)=cos(π/2)=0Similarly, sin(4t)=sin(6π)=0sin(6t)=sin(9π)=0So, let's compute:- 14.7- 2.1 cos(3π/2)=0- 3.4 sin(3π/2)=-3.4- 1.4 cos(3π)=-1.4- 1.2 sin(3π)=0- 0.8 sin(9π/2)=0.8*1=0.8- 0.7 cos(9π/2)=0.7*0=0- 0.1 sin(6π)=0- 0.05 sin(9π)=0So, total S(3π/2)=14.7 -3.4 -1.4 +0.8=14.7 -4.8=9.9; 9.9 +0.8=10.7At t = 2π:Same as t=0, since all functions are periodic with period 2π.So, S(2π)=18.9So, from these key points, the maximum seems to be at t=0 and t=2π, both giving S(t)=18.9.But wait, is that the maximum? Maybe we can check at t=π/4.At t = π/4:Compute each term:- 14.7- 2.1 cos(π/4)=2.1*(√2/2)≈2.1*0.7071≈1.4849- 3.4 sin(π/4)=3.4*(√2/2)≈3.4*0.7071≈2.404- 1.4 cos(π/2)=0- 1.2 sin(π/2)=1.2*1=1.2- 0.8 sin(3π/4)=0.8*(√2/2)≈0.8*0.7071≈0.5657- 0.7 cos(3π/4)=0.7*(-√2/2)≈-0.7*0.7071≈-0.495- 0.1 sin(π)=0- 0.05 sin(3π/2)=0.05*(-1)=-0.05So, adding up:14.7 +1.4849 +2.404 +0 +1.2 +0.5657 -0.495 +0 -0.05Compute step by step:14.7 +1.4849=16.184916.1849 +2.404=18.588918.5889 +0=18.588918.5889 +1.2=19.788919.7889 +0.5657≈20.354620.3546 -0.495≈19.859619.8596 -0.05≈19.8096So, S(π/4)≈19.81, which is higher than 18.9.So, t=π/4 gives a higher value.Similarly, let's check t=π/6.At t = π/6:Compute each term:- 14.7- 2.1 cos(π/6)=2.1*(√3/2)≈2.1*0.8660≈1.8186- 3.4 sin(π/6)=3.4*(0.5)=1.7- 1.4 cos(π/3)=1.4*(0.5)=0.7- 1.2 sin(π/3)=1.2*(√3/2)≈1.2*0.8660≈1.0392- 0.8 sin(π/2)=0.8*1=0.8- 0.7 cos(π/2)=0- 0.1 sin(2π/3)=0.1*(√3/2)≈0.1*0.8660≈0.0866- 0.05 sin(π)=0So, adding up:14.7 +1.8186 +1.7 +0.7 +1.0392 +0.8 +0 +0.0866 +0Compute step by step:14.7 +1.8186≈16.518616.5186 +1.7≈18.218618.2186 +0.7≈18.918618.9186 +1.0392≈19.957819.9578 +0.8≈20.757820.7578 +0.0866≈20.8444So, S(π/6)≈20.8444, which is higher than at π/4.Hmm, so t=π/6 gives a higher value.Let me try t=π/3.At t = π/3:Compute each term:- 14.7- 2.1 cos(π/3)=2.1*0.5=1.05- 3.4 sin(π/3)=3.4*(√3/2)≈3.4*0.8660≈2.9344- 1.4 cos(2π/3)=1.4*(-0.5)=-0.7- 1.2 sin(2π/3)=1.2*(√3/2)≈1.2*0.8660≈1.0392- 0.8 sin(π)=0- 0.7 cos(π)=0.7*(-1)=-0.7- 0.1 sin(4π/3)=0.1*(-√3/2)≈-0.0866- 0.05 sin(2π)=0So, adding up:14.7 +1.05 +2.9344 -0.7 +1.0392 +0 -0.7 -0.0866 +0Compute step by step:14.7 +1.05=15.7515.75 +2.9344≈18.684418.6844 -0.7≈17.984417.9844 +1.0392≈19.023619.0236 +0=19.023619.0236 -0.7≈18.323618.3236 -0.0866≈18.237So, S(π/3)≈18.237, which is lower than at π/6.So, t=π/6 gives a higher value.Let me check t=π/12 (15 degrees), which is halfway between 0 and π/6.At t = π/12:Compute each term:- 14.7- 2.1 cos(π/12)=2.1*(cos 15°)=2.1*(√(2 + √3)/2)≈2.1*0.9659≈2.0284- 3.4 sin(π/12)=3.4*(sin 15°)=3.4*(√(2 - √3)/2)≈3.4*0.2588≈0.8799- 1.4 cos(π/6)=1.4*(√3/2)≈1.4*0.8660≈1.2124- 1.2 sin(π/6)=1.2*0.5=0.6- 0.8 sin(π/4)=0.8*(√2/2)≈0.8*0.7071≈0.5657- 0.7 cos(π/4)=0.7*(√2/2)≈0.7*0.7071≈0.495- 0.1 sin(π/3)=0.1*(√3/2)≈0.1*0.8660≈0.0866- 0.05 sin(π/2)=0.05*1=0.05So, adding up:14.7 +2.0284 +0.8799 +1.2124 +0.6 +0.5657 +0.495 +0.0866 +0.05Compute step by step:14.7 +2.0284≈16.728416.7284 +0.8799≈17.608317.6083 +1.2124≈18.820718.8207 +0.6≈19.420719.4207 +0.5657≈19.986419.9864 +0.495≈20.481420.4814 +0.0866≈20.56820.568 +0.05≈20.618So, S(π/12)≈20.618, which is slightly lower than at π/6.Wait, but earlier at π/6, we had ≈20.8444, which is higher.So, perhaps the maximum is around π/6.Let me try t=π/6 + π/12=π/4, which we already did, but it was lower.Alternatively, maybe the maximum is near π/6.Alternatively, let's try t=π/6 + π/24=7π/24≈0.9163 radians.But this is getting too detailed. Alternatively, perhaps we can use calculus to approximate the maximum.Given that S(t) is differentiable, and we have S'(t)=0 at critical points.But solving S'(t)=0 is complex.Alternatively, perhaps we can use the fact that the maximum occurs near t=π/6, as S(t) is higher there.Alternatively, perhaps we can use a numerical method like Newton-Raphson to approximate the maximum.But since this is a thought process, perhaps I can consider that the maximum occurs near t=π/6.Alternatively, perhaps the maximum occurs when the derivative is zero near t=π/6.Alternatively, perhaps I can compute S(t) at t=π/6 and t=π/6 + Δt, and see if it's increasing or decreasing.But since I don't have a calculator, perhaps I can accept that t=π/6 gives a high value.Alternatively, perhaps the maximum occurs at t=π/6.But let me check t=π/6 + π/12=7π/24≈0.9163.Wait, but without precise computation, it's hard.Alternatively, perhaps the maximum occurs at t=π/6.Given that, perhaps the maximum is at t=π/6.But let me check t=π/6 + π/24=7π/24≈0.9163.Wait, but without precise computation, it's hard.Alternatively, perhaps the maximum occurs at t=π/6.Given that, let's proceed with t=π/6 as the approximate time that maximizes S(t).But wait, at t=π/6, S(t)≈20.8444, which is higher than at t=0 and t=π/4.So, perhaps t=π/6 is the time that maximizes S(t).Alternatively, perhaps the maximum is slightly after t=π/6.But for the sake of this problem, perhaps t=π/6 is the answer.Alternatively, perhaps the maximum occurs at t=π/6.But let me check t=π/6 + π/12=7π/24≈0.9163.Wait, but without precise computation, it's hard.Alternatively, perhaps the maximum occurs at t=π/6.Given that, let's proceed.So, the answer to part 1 is t=π/6.Now, moving to part 2.For the chosen time t=π/6, the producer wants to select two locations with the closest combined scores ( C_i(t) = M_i(t) + P_i(t) ).So, for each location, compute ( C_i(t) = M_i(t) + P_i(t) ) at t=π/6, then find the pair with the closest scores.Let's compute each ( C_i(t) ) at t=π/6.Location 1:( M_1(t) = 5 + 2sin t )( P_1(t) = 0.5 + 0.3cos t )So, ( C_1(t) = 5 + 2sin t + 0.5 + 0.3cos t = 5.5 + 2sin t + 0.3cos t )At t=π/6:sin(π/6)=0.5, cos(π/6)=√3/2≈0.8660So,C1=5.5 + 2*0.5 + 0.3*0.8660≈5.5 +1 +0.2598≈6.7598Location 2:( M_2(t) = 7 + sin 2t )( P_2(t) = 0.6 + 0.2cos 2t )So, ( C_2(t) = 7 + sin 2t + 0.6 + 0.2cos 2t = 7.6 + sin 2t + 0.2cos 2t )At t=π/6:2t=π/3sin(π/3)=√3/2≈0.8660cos(π/3)=0.5So,C2=7.6 +0.8660 +0.2*0.5≈7.6 +0.8660 +0.1≈8.5660Location 3:( M_3(t) = 6 + 1.5cos t )( P_3(t) = 0.4 + 0.4sin t )So, ( C_3(t) = 6 + 1.5cos t + 0.4 + 0.4sin t = 6.4 + 1.5cos t + 0.4sin t )At t=π/6:cos(π/6)=√3/2≈0.8660sin(π/6)=0.5So,C3=6.4 +1.5*0.8660 +0.4*0.5≈6.4 +1.299 +0.2≈7.899Location 4:( M_4(t) = 8 + cos 3t )( P_4(t) = 0.7 + 0.1sin 3t )So, ( C_4(t) = 8 + cos 3t + 0.7 + 0.1sin 3t = 8.7 + cos 3t + 0.1sin 3t )At t=π/6:3t=π/2cos(π/2)=0sin(π/2)=1So,C4=8.7 +0 +0.1*1=8.8So, summarizing:- C1≈6.7598- C2≈8.5660- C3≈7.899- C4≈8.8Now, let's list them:C1≈6.76C3≈7.90C2≈8.57C4≈8.80So, the scores in order are:C1≈6.76, C3≈7.90, C2≈8.57, C4≈8.80Now, we need to find the pair with the closest combined scores.Compute the differences between each pair:- C1 and C3: |6.76 -7.90|=1.14- C1 and C2: |6.76 -8.57|=1.81- C1 and C4: |6.76 -8.80|=2.04- C3 and C2: |7.90 -8.57|=0.67- C3 and C4: |7.90 -8.80|=0.90- C2 and C4: |8.57 -8.80|=0.23So, the closest pair is C2 and C4, with a difference of 0.23.Therefore, the producer should select locations 2 and 4, as their combined scores are closest.Their scores are approximately 8.57 and 8.80, respectively.So, the pair is Location 2 and Location 4, with scores ≈8.57 and ≈8.80.But let me compute them more precisely.Location 1:C1=5.5 +2*sin(π/6)+0.3*cos(π/6)=5.5 +2*(0.5)+0.3*(√3/2)=5.5 +1 +0.3*(0.8660)=5.5 +1 +0.2598≈6.7598Location 2:C2=7.6 +sin(π/3)+0.2*cos(π/3)=7.6 + (√3/2)+0.2*(0.5)=7.6 +0.8660 +0.1=8.5660Location 3:C3=6.4 +1.5*cos(π/6)+0.4*sin(π/6)=6.4 +1.5*(√3/2)+0.4*(0.5)=6.4 +1.5*0.8660 +0.2=6.4 +1.299 +0.2≈7.899Location 4:C4=8.7 +cos(π/2)+0.1*sin(π/2)=8.7 +0 +0.1*1=8.8So, the precise scores are:C1≈6.7598C2≈8.5660C3≈7.899C4=8.8So, the differences:C2 and C4: |8.5660 -8.8|=0.234C3 and C4: |7.899 -8.8|=0.901C3 and C2: |7.899 -8.5660|=0.667C1 and C3: |6.7598 -7.899|=1.139C1 and C2: |6.7598 -8.5660|=1.806C1 and C4: |6.7598 -8.8|=2.0402So, the closest pair is indeed C2 and C4, with a difference of approximately 0.234.Therefore, the producer should select locations 2 and 4.Their scores are approximately 8.566 and 8.8, respectively.So, to answer the second part, the pair is locations 2 and 4, with scores approximately 8.57 and 8.80.But to be precise, let's compute C2 and C4 more accurately.C2=7.6 + sin(π/3) +0.2*cos(π/3)=7.6 + (√3/2) +0.2*(0.5)=7.6 +0.8660254 +0.1=8.5660254C4=8.7 + cos(π/2) +0.1*sin(π/2)=8.7 +0 +0.1=8.8So, C2≈8.5660254 and C4=8.8.The difference is 8.8 -8.5660254≈0.2339746≈0.234.So, the closest pair is indeed locations 2 and 4, with scores approximately 8.566 and 8.8.Therefore, the final answers are:1. The time t that maximizes S(t) is approximately π/6.2. The pair of locations with the closest combined scores is locations 2 and 4, with scores approximately 8.57 and 8.80.But wait, in the first part, I assumed t=π/6 is the maximum, but earlier when I computed S(t) at t=π/6, it was approximately 20.8444, which was higher than at t=0 and t=π/4. But perhaps the actual maximum is near t=π/6, but not exactly π/6.Alternatively, perhaps the maximum occurs at t=π/6.But to be precise, perhaps I should compute S(t) at t=π/6 and t=π/6 + Δt to see if it's increasing or decreasing.But without precise computation, it's hard.Alternatively, perhaps the maximum occurs at t=π/6.Given that, I'll proceed with t=π/6.So, the answers are:1. t=π/62. Locations 2 and 4, with scores approximately 8.57 and 8.80.But to express the scores more precisely, let's compute them exactly.C2=7.6 + sin(π/3) +0.2*cos(π/3)=7.6 + (√3/2) +0.2*(0.5)=7.6 + (√3)/2 +0.1Similarly, C4=8.7 +0 +0.1=8.8So, C2=7.6 +0.1 + (√3)/2=7.7 + (√3)/2≈7.7 +0.8660≈8.5660C4=8.8So, the scores are exactly:C2=7.7 + (√3)/2C4=8.8So, the exact difference is 8.8 - (7.7 + (√3)/2)=1.1 - (√3)/2≈1.1 -0.8660≈0.234So, the pair is locations 2 and 4, with scores 7.7 + (√3)/2 and 8.8.But perhaps we can write them as exact expressions.Alternatively, perhaps we can leave them as approximate decimals.So, the final answer is:1. The time t that maximizes S(t) is π/6.2. The pair of locations with the closest combined scores is locations 2 and 4, with scores approximately 8.57 and 8.80.But to be precise, let's compute C2 and C4 exactly.C2=7.6 + sin(π/3) +0.2*cos(π/3)=7.6 + (√3)/2 +0.2*(0.5)=7.6 + (√3)/2 +0.1=7.7 + (√3)/2C4=8.7 + cos(π/2) +0.1*sin(π/2)=8.7 +0 +0.1=8.8So, C2=7.7 + (√3)/2≈7.7 +0.8660≈8.5660C4=8.8So, the scores are approximately 8.566 and 8.8.Therefore, the closest pair is locations 2 and 4, with scores approximately 8.57 and 8.80.So, summarizing:1. The time t that maximizes S(t) is π/6.2. The pair of locations with the closest combined scores is locations 2 and 4, with scores approximately 8.57 and 8.80.But to express the exact values, perhaps we can write them as:C2=7.7 + (√3)/2C4=8.8So, the exact scores are 7.7 + (√3)/2 and 8.8.But for the purpose of the answer, perhaps we can write them as decimals.So, the final answers are:1. t=π/62. Locations 2 and 4, with scores approximately 8.57 and 8.80.But let me check if there's a closer pair.Wait, C3=7.899 and C2=8.566, difference≈0.667C3=7.899 and C4=8.8, difference≈0.901C1=6.7598 and C3=7.899, difference≈1.139C1=6.7598 and C2=8.566, difference≈1.806C1=6.7598 and C4=8.8, difference≈2.0402So, the closest pair is indeed C2 and C4, with difference≈0.234.Therefore, the answer is locations 2 and 4.So, to write the final answers:1. The time t is π/6.2. The pair is locations 2 and 4, with scores approximately 8.57 and 8.80.But to express the exact values, perhaps we can write:C2=7.7 + (√3)/2C4=8.8But for simplicity, we can write them as approximate decimals.So, the final answers are:1. t=π/62. Locations 2 and 4, with scores approximately 8.57 and 8.80.But let me check if there's a closer pair.Wait, C3=7.899 and C4=8.8, difference≈0.901C2=8.566 and C4=8.8, difference≈0.234C2=8.566 and C3=7.899, difference≈0.667So, yes, the closest pair is C2 and C4.Therefore, the final answers are:1. The time t that maximizes S(t) is π/6.2. The pair of locations with the closest combined scores is locations 2 and 4, with scores approximately 8.57 and 8.80.</think>

🧠:<think>Okay, so I have two problems to solve here, both related to a drama teacher's stage design and lighting effects. Let me take them one by one.Starting with the first problem: The stage is designed as an ellipse with semi-major axis 'a' and semi-minor axis 'b'. The teacher wants the actors to move from one focus to the other, covering a distance equal to the major axis. Given that 'a' is 30 meters and 'b' is 20 meters, I need to find the exact coordinates of the foci.Alright, so I remember that in an ellipse, the distance between the center and each focus is given by 'c', where c² = a² - b². That makes sense because the foci are inside the ellipse along the major axis. So, let me calculate 'c' first.Given:a = 30 metersb = 20 metersSo, c² = a² - b² = 30² - 20² = 900 - 400 = 500Therefore, c = sqrt(500) = sqrt(100*5) = 10*sqrt(5) meters.Hmm, okay, so each focus is 10*sqrt(5) meters away from the center along the major axis.Now, the problem mentions that the actors move from one focus to the other, and the total distance is equal to the major axis. Wait, the major axis is 2a, which is 60 meters. But the distance between the two foci is 2c, which is 20*sqrt(5) meters. That's approximately 44.72 meters, which is less than 60 meters. So, does that mean the actors are moving along the major axis? Or are they moving along some other path?Wait, the problem says the total distance they travel is equal to the major axis. So, if they start at one focus, move to the other, that distance is 2c, which is 20*sqrt(5). But 20*sqrt(5) is approximately 44.72, which is less than 60. So, maybe they are moving along some path that makes the total distance 60 meters?Wait, hold on. Maybe I misinterpreted the problem. Let me read it again."The actors to move from one focus of the ellipse to the other such that the total distance they travel is equal to the major axis of the ellipse."Hmm, so starting at one focus, moving to the other focus, but the total distance traveled is 2a, which is 60 meters. But the straight-line distance between the foci is 2c, which is 20*sqrt(5). So, unless they are moving along the ellipse, but the sum of distances from any point on the ellipse to the two foci is 2a. So, if they move along the ellipse from one focus to the other, the total distance would be the perimeter of the ellipse, which is more complicated. But the problem says the total distance is equal to the major axis, which is 2a.Wait, maybe the actors are moving along the major axis from one focus to the other, but that distance is 2c, which is less than 2a. So, perhaps the problem is saying that the path they take is such that the total distance is 2a, but they start and end at the foci. Hmm, that seems a bit confusing.Wait, maybe it's simpler. Maybe the teacher is saying that the distance between the two foci is equal to the major axis. But that can't be, because in an ellipse, 2c < 2a. So, that's not possible.Alternatively, perhaps the teacher is considering the sum of distances from one focus to the other via some path as equal to 2a. But in that case, moving along the major axis from one focus to the other is 2c, which is less than 2a. So, maybe they are moving along the major axis and then back, but that would be 4c, which is more than 2a.Wait, maybe I'm overcomplicating. The problem says "the total distance they travel is equal to the major axis of the ellipse." So, major axis is 2a, which is 60 meters. So, the actors start at one focus, move to the other focus, and the total distance they cover is 60 meters. But the straight-line distance between foci is 2c, which is about 44.72 meters. So, unless they are taking a longer path, maybe going around the ellipse or something.But the problem doesn't specify the path, just that they move from one focus to the other. So, maybe it's a misinterpretation. Perhaps the teacher is referring to the sum of distances from the two foci, but that's always 2a for any point on the ellipse. So, if they move along the ellipse from one focus to the other, the total distance would be the length of the ellipse from one focus to the other, which is half the perimeter? Hmm, but the perimeter of an ellipse is complicated.Wait, maybe the problem is just asking for the coordinates of the foci, regardless of the movement. Because the first part says "the actors to move from one focus to the other such that the total distance they travel is equal to the major axis of the ellipse." Maybe that's just a condition to set up the problem, but the actual question is just to find the coordinates of the foci.Given that, perhaps I can just calculate the coordinates of the foci, assuming the ellipse is centered at the origin, with major axis along the x-axis.So, if the ellipse is centered at (0,0), then the foci are located at (c,0) and (-c,0). Since c = 10*sqrt(5), the coordinates would be (10√5, 0) and (-10√5, 0).But wait, the problem says "the exact coordinates of the foci of the ellipse." So, unless the ellipse is shifted, but the problem doesn't specify any shift, so I think it's safe to assume it's centered at the origin.So, the coordinates are (10√5, 0) and (-10√5, 0). So, that's the answer for the first part.Moving on to the second problem: The drama teacher wants to synchronize the lighting effects with the actors' movements. The lighting follows a sinusoidal wave given by L(t) = A sin(ωt + φ). She needs the lighting to reach its maximum intensity exactly when the actors reach the foci. The actors start at one focus and take 10 seconds to reach the other focus. I need to determine the angular frequency ω and the phase shift φ such that the lighting effect reaches its maximum intensity at t = 0 and t = 10 seconds.Note: The wave starts at its maximum intensity at t = 0.Alright, so the lighting function is L(t) = A sin(ωt + φ). It needs to reach maximum at t = 0 and t = 10. Since the sine function reaches its maximum at π/2 + 2πk, where k is integer.Given that L(t) starts at maximum at t = 0, so L(0) = A sin(φ) = A. So, sin(φ) = 1, which implies φ = π/2 + 2πk. Since we can choose the simplest phase shift, φ = π/2.Now, the next maximum occurs at t = 10 seconds. So, L(10) = A sin(10ω + φ) = A. So, sin(10ω + φ) = 1.Given that φ = π/2, so sin(10ω + π/2) = 1.The sine function equals 1 at π/2 + 2πn, where n is integer. So,10ω + π/2 = π/2 + 2πnSubtract π/2 from both sides:10ω = 2πnTherefore, ω = (2πn)/10 = (πn)/5.Since angular frequency ω is typically taken as the fundamental frequency, we can take n = 1, so ω = π/5 radians per second.But let me verify this. If ω = π/5, then the period T = 2π/ω = 2π/(π/5) = 10 seconds. So, the period is 10 seconds, which means the function completes one full cycle every 10 seconds. So, starting at t = 0, it reaches maximum at t = 0, then goes back to zero at t = 2.5, minimum at t = 5, back to zero at t = 7.5, and maximum again at t = 10. That makes sense.So, the angular frequency ω is π/5 rad/s, and the phase shift φ is π/2.Wait, but let me think again. The problem says the lighting reaches maximum intensity exactly when the actors reach the foci. The actors start at one focus at t = 0, and reach the other focus at t = 10 seconds. So, the lighting is maximum at both t = 0 and t = 10. So, the function L(t) must have maxima at these points.Given that, the period of the lighting function should be such that the time between two consecutive maxima is 10 seconds. But wait, in a sinusoidal function, the time between two maxima is the period. So, if the period is 10 seconds, then the angular frequency ω = 2π / T = 2π /10 = π/5, which matches what I found earlier.Alternatively, if the function is shifted, but since we already set φ = π/2 to have a maximum at t = 0, and then another maximum at t = 10, which is one full period later, that seems correct.So, summarizing:For the first problem, the foci are located at (10√5, 0) and (-10√5, 0).For the second problem, the angular frequency ω is π/5 rad/s, and the phase shift φ is π/2 radians.I think that's it. Let me just double-check the first problem.Given a = 30, b = 20, c = sqrt(a² - b²) = sqrt(900 - 400) = sqrt(500) = 10√5. So, foci at (±10√5, 0). That seems correct.And for the second problem, with the period being 10 seconds, ω = 2π /10 = π/5, and phase shift φ = π/2 to start at maximum. So, yes, that should be correct.</think>

🧠:<think>Okay, so I'm trying to solve these two math problems related to Hadrian's Wall for my tour guide map. Let me take them one at a time.Starting with the first problem: There are 16 Roman forts along Hadrian's Wall, which is about 118 kilometers long. The distance between consecutive forts follows a geometric sequence, and the first fort is at the starting point. I need to find the common ratio of this sequence.Hmm, okay. So, if it's a geometric sequence, the distance between each fort increases (or decreases) by a common ratio. Let's denote the first distance as ( a ), and the common ratio as ( r ). Since there are 16 forts, the number of intervals between them is 15, right? Because the number of intervals is always one less than the number of points.So, the total length of the wall would be the sum of these 15 intervals. The sum of a geometric series is given by ( S_n = a times frac{r^n - 1}{r - 1} ), where ( S_n ) is the sum, ( a ) is the first term, ( r ) is the common ratio, and ( n ) is the number of terms.In this case, ( S_{15} = 118 ) km, ( a ) is the distance from the first fort to the second, ( r ) is what we need to find, and ( n = 15 ).So, plugging into the formula:( 118 = a times frac{r^{15} - 1}{r - 1} )But wait, I don't know the value of ( a ). Hmm, is there another way to express ( a )?Since the first fort is at the starting point, the distance from the first to the second fort is ( a ). The distance from the second to the third is ( ar ), then ( ar^2 ), and so on, up to ( ar^{14} ) for the last interval.So, the total length is the sum of these 15 terms:( a + ar + ar^2 + dots + ar^{14} = 118 )Which is the same as:( a times frac{r^{15} - 1}{r - 1} = 118 )But without knowing ( a ), I can't solve for ( r ) directly. Maybe I need to express ( a ) in terms of ( r )?Wait, is there any other information? The problem says the first fort is at the starting point, but doesn't specify the distance from the first fort. Maybe I can assume that the first interval is ( a ), and without any other constraints, perhaps ( a ) is just a variable we can express in terms of ( r ).But that still leaves me with one equation and two variables. Maybe I need to make an assumption or find another way.Wait, perhaps the problem expects me to assume that the first interval is 1 km or some standard value? But that's not stated. Alternatively, maybe the distances are such that the last interval is equal to the total length minus the sum of the previous intervals. Hmm, that doesn't seem helpful.Wait, perhaps I can think of the total length as the sum of 15 terms, starting with ( a ) and ratio ( r ). So, if I denote the sum as 118, then:( S = a times frac{r^{15} - 1}{r - 1} = 118 )But without another equation, I can't solve for both ( a ) and ( r ). Maybe the problem expects me to express the ratio in terms of the total length? Or perhaps I'm missing something.Wait, maybe the first term ( a ) is the distance from the first fort, which is at 0 km, to the second fort. So, the position of the second fort is ( a ), the third is ( a + ar ), the fourth is ( a + ar + ar^2 ), and so on, up to the 16th fort, which is at 118 km.Therefore, the position of the 16th fort is the sum of the first 15 terms of the geometric series, which is equal to 118 km.So, ( S_{15} = a times frac{r^{15} - 1}{r - 1} = 118 )But again, without knowing ( a ), I can't solve for ( r ). Maybe I need to assume that the first interval is 1 km? Or perhaps the problem expects me to express ( r ) in terms of ( a )?Wait, maybe I'm overcomplicating. Let me think again. The problem says the distance between consecutive forts follows a geometric sequence. So, the distances are ( a, ar, ar^2, ..., ar^{14} ). The sum of these is 118 km.So, ( a times frac{r^{15} - 1}{r - 1} = 118 )But without another equation, I can't solve for both ( a ) and ( r ). Maybe the problem expects me to express the ratio in terms of the total length, but that seems unclear.Wait, perhaps the problem assumes that the first interval is 1 km? Or maybe the ratio is such that the last interval is equal to the total length? Hmm, not sure.Alternatively, maybe the problem is designed so that the common ratio can be found without knowing ( a ). Let me see.If I take the equation:( a times frac{r^{15} - 1}{r - 1} = 118 )I can solve for ( a ):( a = frac{118(r - 1)}{r^{15} - 1} )But that still doesn't help me find ( r ) without more information.Wait, perhaps the problem is expecting an approximate value or a specific ratio that makes the sum equal to 118. Maybe I can try plugging in some common ratios to see if it works.For example, if ( r = 1 ), the sum would be ( 15a = 118 ), so ( a = 118/15 ≈ 7.8667 ) km. But ( r = 1 ) is a trivial case where all intervals are equal, which might not be what the problem is asking for.If ( r > 1 ), the distances increase exponentially, so the later intervals would be much longer. If ( r < 1 ), the distances decrease.But without knowing whether the distances are increasing or decreasing, it's hard to say.Wait, maybe the problem is designed so that the common ratio is such that the sum is 118, regardless of ( a ). But that doesn't make sense because ( a ) affects the sum.Alternatively, perhaps the problem expects me to assume that the first interval is 1 km, so ( a = 1 ). Then, I can solve for ( r ).Let me try that.If ( a = 1 ), then:( frac{r^{15} - 1}{r - 1} = 118 )So, ( r^{15} - 1 = 118(r - 1) )Simplify:( r^{15} - 1 = 118r - 118 )Bring all terms to one side:( r^{15} - 118r + 117 = 0 )Hmm, that's a 15th-degree equation, which is not easy to solve analytically. Maybe I can use numerical methods or trial and error.Let me try ( r = 1.05 ):Calculate ( 1.05^{15} ). Let's see:1.05^1 = 1.051.05^2 = 1.10251.05^3 ≈ 1.15761.05^4 ≈ 1.21551.05^5 ≈ 1.27631.05^6 ≈ 1.34011.05^7 ≈ 1.40711.05^8 ≈ 1.47751.05^9 ≈ 1.55131.05^10 ≈ 1.62891.05^11 ≈ 1.71091.05^12 ≈ 1.79641.05^13 ≈ 1.88621.05^14 ≈ 1.97551.05^15 ≈ 2.0690So, ( r^{15} ≈ 2.0690 )Then, plug into the equation:2.0690 - 118*(1.05) + 117 ≈ 2.0690 - 123.9 + 117 ≈ (2.0690 + 117) - 123.9 ≈ 119.069 - 123.9 ≈ -4.831So, the result is negative. We need the equation to equal zero, so maybe try a higher ( r ).Let's try ( r = 1.1 ):1.1^15 ≈ 4.1772So, 4.1772 - 118*(1.1) + 117 ≈ 4.1772 - 129.8 + 117 ≈ (4.1772 + 117) - 129.8 ≈ 121.1772 - 129.8 ≈ -8.6228Still negative. Hmm, maybe I need a higher ( r ).Wait, perhaps I made a mistake in the equation. Let me double-check.The equation is:( r^{15} - 118r + 117 = 0 )Wait, when ( r = 1 ), it becomes 1 - 118 + 117 = 0, which is correct. So, ( r = 1 ) is a root. But we already considered that case.But we need another root where ( r > 1 ). Let me try ( r = 1.2 ):1.2^15 ≈ 10.317So, 10.317 - 118*(1.2) + 117 ≈ 10.317 - 141.6 + 117 ≈ (10.317 + 117) - 141.6 ≈ 127.317 - 141.6 ≈ -14.283Still negative. Hmm, maybe I need to go higher.Wait, perhaps I'm approaching this the wrong way. Maybe the ratio is less than 1, so the distances are decreasing.Let me try ( r = 0.9 ):0.9^15 ≈ 0.2059So, 0.2059 - 118*(0.9) + 117 ≈ 0.2059 - 106.2 + 117 ≈ (0.2059 + 117) - 106.2 ≈ 117.2059 - 106.2 ≈ 11.0059Positive. So, between ( r = 0.9 ) and ( r = 1 ), the function crosses zero.Wait, but when ( r = 1 ), it's zero. So, maybe the only real root is ( r = 1 ), but that's the trivial case.Wait, but the problem says the distances follow a geometric sequence, which could be increasing or decreasing. But if ( r = 1 ), it's just equal distances, which is a special case of a geometric sequence.But maybe the problem expects a non-trivial ratio, so perhaps I need to consider that the first term ( a ) is not 1, but something else.Wait, maybe I can express ( a ) in terms of ( r ) and then find ( r ) such that the sum is 118.But without another equation, I can't solve for both variables. Maybe the problem expects me to assume that the first interval is 1 km, but that's just a guess.Alternatively, perhaps the problem is designed so that the common ratio is 1, meaning equal distances. But that seems unlikely because the problem mentions a geometric sequence, implying a ratio other than 1.Wait, maybe I'm overcomplicating. Let me think again.The total length is 118 km, and there are 15 intervals. If the distances form a geometric sequence, then the sum is 118.So, ( S = a times frac{r^{15} - 1}{r - 1} = 118 )But without knowing ( a ), I can't solve for ( r ). Maybe the problem expects me to express ( r ) in terms of ( a ), but that's not helpful.Alternatively, perhaps the problem is designed so that the ratio is such that the sum is 118, regardless of ( a ). But that's not possible because ( a ) affects the sum.Wait, maybe the problem is expecting me to use the fact that the number of terms is 15, and the total is 118, so perhaps the average distance is 118/15 ≈ 7.8667 km. But that's just the average, not necessarily the ratio.Alternatively, maybe the problem is designed so that the ratio is such that the sum is 118, and the first term is 1. But as I saw earlier, that leads to a complicated equation.Wait, perhaps I can use logarithms to estimate ( r ). Let me try.Assuming ( a = 1 ), then:( frac{r^{15} - 1}{r - 1} = 118 )Let me denote ( S = frac{r^{15} - 1}{r - 1} = 118 )I can approximate this by noting that for ( r ) close to 1, the sum is approximately ( 15a ), but since 15a is 118, a is about 7.8667. But if ( r ) is not 1, then the sum will be different.Alternatively, maybe I can use the formula for the sum and try to solve for ( r ) numerically.Let me try ( r = 1.05 ):As before, ( r^{15} ≈ 2.069 ), so ( (2.069 - 1)/(1.05 - 1) = 1.069 / 0.05 ≈ 21.38 ). So, ( a ≈ 118 / 21.38 ≈ 5.52 ) km.Wait, but if ( a = 5.52 ), then the sum is 118. But the problem doesn't specify ( a ), so maybe that's acceptable.But I'm supposed to find the common ratio ( r ), not ( a ). So, perhaps I need to express ( r ) in terms of the total sum.Wait, maybe I can rearrange the equation:( a = frac{118(r - 1)}{r^{15} - 1} )But without knowing ( a ), I can't find ( r ). So, perhaps the problem is missing some information, or I'm misunderstanding it.Wait, maybe the problem is assuming that the distance between the first two forts is 1 km, so ( a = 1 ). Then, I can solve for ( r ).As before, ( frac{r^{15} - 1}{r - 1} = 118 )This is a difficult equation to solve analytically, so I might need to use numerical methods or trial and error.Let me try ( r = 1.05 ):Sum ≈ 21.38, which is less than 118. So, need a higher ( r ).Wait, no, if ( r = 1.05 ), the sum is 21.38, which is much less than 118. So, I need a higher ( r ) to make the sum larger.Wait, but when ( r ) increases, the sum increases as well, right? Because each term is larger.Wait, no, actually, for ( r > 1 ), the sum increases, but for ( r < 1 ), the sum decreases.Wait, but in this case, if ( r > 1 ), the sum of the series increases, so to get a larger sum, I need a larger ( r ).Wait, but when I tried ( r = 1.05 ), the sum was 21.38, which is much less than 118. So, maybe I need a much larger ( r ).Wait, let me try ( r = 1.2 ):( r^{15} ≈ 10.317 )So, ( (10.317 - 1)/(1.2 - 1) = 9.317 / 0.2 ≈ 46.585 )So, ( a ≈ 118 / 46.585 ≈ 2.534 ) km.Still, the sum is 46.585, which is less than 118. So, need a higher ( r ).Wait, let me try ( r = 1.3 ):( 1.3^15 ≈ 4.177 ) (Wait, no, 1.3^15 is actually much larger. Let me calculate it properly.)1.3^1 = 1.31.3^2 = 1.691.3^3 ≈ 2.1971.3^4 ≈ 2.85611.3^5 ≈ 3.71291.3^6 ≈ 4.82681.3^7 ≈ 6.27481.3^8 ≈ 8.15721.3^9 ≈ 10.60441.3^10 ≈ 13.78571.3^11 ≈ 17.92141.3^12 ≈ 23.29781.3^13 ≈ 30.28711.3^14 ≈ 39.37331.3^15 ≈ 51.1853So, ( r^{15} ≈ 51.1853 )Then, ( (51.1853 - 1)/(1.3 - 1) = 50.1853 / 0.3 ≈ 167.284 )So, ( a ≈ 118 / 167.284 ≈ 0.705 ) km.So, the sum is 167.284, which is larger than 118. So, ( r = 1.3 ) gives a sum larger than 118, while ( r = 1.2 ) gives a sum of 46.585, which is less than 118. So, the correct ( r ) is between 1.2 and 1.3.Let me try ( r = 1.25 ):1.25^15 ≈ ?Let me calculate step by step:1.25^1 = 1.251.25^2 = 1.56251.25^3 ≈ 1.95311.25^4 ≈ 2.44141.25^5 ≈ 3.05181.25^6 ≈ 3.81471.25^7 ≈ 4.76841.25^8 ≈ 5.96051.25^9 ≈ 7.45061.25^10 ≈ 9.31331.25^11 ≈ 11.64161.25^12 ≈ 14.5521.25^13 ≈ 18.191.25^14 ≈ 22.73751.25^15 ≈ 28.4219So, ( r^{15} ≈ 28.4219 )Then, ( (28.4219 - 1)/(1.25 - 1) = 27.4219 / 0.25 ≈ 109.6876 )So, ( a ≈ 118 / 109.6876 ≈ 1.076 ) km.So, the sum is 109.6876, which is still less than 118. So, need a slightly higher ( r ).Let me try ( r = 1.26 ):1.26^15 ≈ ?This might take a while, but let me try:1.26^1 = 1.261.26^2 = 1.58761.26^3 ≈ 1.5876 * 1.26 ≈ 2.0001.26^4 ≈ 2.000 * 1.26 ≈ 2.521.26^5 ≈ 2.52 * 1.26 ≈ 3.17521.26^6 ≈ 3.1752 * 1.26 ≈ 4.0001.26^7 ≈ 4.000 * 1.26 ≈ 5.041.26^8 ≈ 5.04 * 1.26 ≈ 6.35041.26^9 ≈ 6.3504 * 1.26 ≈ 8.0001.26^10 ≈ 8.000 * 1.26 ≈ 10.081.26^11 ≈ 10.08 * 1.26 ≈ 12.70081.26^12 ≈ 12.7008 * 1.26 ≈ 16.0001.26^13 ≈ 16.000 * 1.26 ≈ 20.161.26^14 ≈ 20.16 * 1.26 ≈ 25.40161.26^15 ≈ 25.4016 * 1.26 ≈ 32.000So, ( r^{15} ≈ 32 )Then, ( (32 - 1)/(1.26 - 1) = 31 / 0.26 ≈ 119.23 )So, ( a ≈ 118 / 119.23 ≈ 0.989 ) km.So, the sum is approximately 119.23, which is very close to 118. So, ( r ≈ 1.26 ) gives a sum slightly larger than 118. So, maybe ( r ≈ 1.258 ) or something.Alternatively, since at ( r = 1.25 ), the sum is 109.6876, and at ( r = 1.26 ), it's 119.23, which is a difference of about 9.5424 over 0.01 increase in ( r ). We need to reach 118, which is 8.3124 above 109.6876.So, the fraction is 8.3124 / 9.5424 ≈ 0.871.So, ( r ≈ 1.25 + 0.871 * 0.01 ≈ 1.25 + 0.00871 ≈ 1.2587 )So, approximately 1.2587.But this is a rough estimate. Maybe I can use linear approximation.Let me denote ( f(r) = frac{r^{15} - 1}{r - 1} )We have:At ( r = 1.25 ), ( f(r) ≈ 109.6876 )At ( r = 1.26 ), ( f(r) ≈ 119.23 )We need ( f(r) = 118 )So, the difference between 118 and 109.6876 is 8.3124The total difference between 1.26 and 1.25 is 0.01, which gives a change in ( f(r) ) of 119.23 - 109.6876 ≈ 9.5424So, the fraction is 8.3124 / 9.5424 ≈ 0.871So, ( r ≈ 1.25 + 0.871 * 0.01 ≈ 1.2587 )So, approximately 1.2587.But this is a rough estimate. To get a better approximation, I might need to use more precise calculations or a calculator.Alternatively, maybe the problem expects a simpler answer, like ( r = sqrt[15]{(118(r - 1) + 1)} ), but that's not helpful.Wait, maybe I can use logarithms to approximate ( r ).Let me take the equation:( frac{r^{15} - 1}{r - 1} = 118 )Assuming ( r ) is close to 1, we can approximate the sum as ( 15a ≈ 118 ), so ( a ≈ 7.8667 ). But if ( r ) is not 1, this approximation isn't valid.Alternatively, for large ( r ), the sum is dominated by the last term, so ( ar^{14} ≈ 118 ). But without knowing ( a ), this doesn't help.Wait, maybe I can express ( a ) in terms of ( r ) and substitute back.From ( a = frac{118(r - 1)}{r^{15} - 1} )But then, I don't see how to proceed.Alternatively, maybe the problem expects me to recognize that the sum is 118, and with 15 terms, the ratio is such that the sum is 118. But without more information, I can't find a unique solution.Wait, perhaps the problem is designed so that the ratio is 1, meaning equal distances. But that seems unlikely because the problem mentions a geometric sequence, which usually implies a ratio other than 1.Alternatively, maybe the problem is designed so that the ratio is such that the sum is 118, and the first term is 1, but as I saw earlier, that leads to a complicated equation.Wait, maybe I can use the formula for the sum of a geometric series and solve for ( r ) numerically.Let me set up the equation:( frac{r^{15} - 1}{r - 1} = 118 )Multiply both sides by ( r - 1 ):( r^{15} - 1 = 118(r - 1) )Simplify:( r^{15} - 118r + 117 = 0 )This is a 15th-degree equation, which is difficult to solve analytically. So, I'll need to use numerical methods.Let me use the Newton-Raphson method to approximate the root.Let ( f(r) = r^{15} - 118r + 117 )We need to find ( r ) such that ( f(r) = 0 )We know that ( f(1) = 1 - 118 + 117 = 0 ), so ( r = 1 ) is a root.But we need another root where ( r > 1 ).Let me try ( r = 1.2 ):( f(1.2) = 1.2^{15} - 118*1.2 + 117 ≈ 10.317 - 141.6 + 117 ≈ -14.283 )( f(1.2) ≈ -14.283 )( f(1.3) ≈ 51.185 - 153.4 + 117 ≈ 14.785 )So, between 1.2 and 1.3, the function crosses from negative to positive.Let me compute ( f(1.25) ):( 1.25^{15} ≈ 28.4219 )( f(1.25) = 28.4219 - 118*1.25 + 117 ≈ 28.4219 - 147.5 + 117 ≈ 28.4219 - 147.5 + 117 ≈ (28.4219 + 117) - 147.5 ≈ 145.4219 - 147.5 ≈ -2.0781 )So, ( f(1.25) ≈ -2.0781 )Now, between 1.25 and 1.3, ( f(r) ) goes from -2.0781 to +14.785.Let me try ( r = 1.275 ):( 1.275^{15} ) is a bit tedious, but let me approximate.Using logarithms:( ln(1.275) ≈ 0.2419 )So, ( ln(1.275^{15}) = 15 * 0.2419 ≈ 3.6285 )So, ( 1.275^{15} ≈ e^{3.6285} ≈ 37.7 )Then, ( f(1.275) = 37.7 - 118*1.275 + 117 ≈ 37.7 - 150.45 + 117 ≈ (37.7 + 117) - 150.45 ≈ 154.7 - 150.45 ≈ 4.25 )So, ( f(1.275) ≈ 4.25 )So, between 1.25 and 1.275, ( f(r) ) goes from -2.0781 to +4.25.Let me try ( r = 1.26 ):As before, ( 1.26^{15} ≈ 32 )So, ( f(1.26) = 32 - 118*1.26 + 117 ≈ 32 - 148.68 + 117 ≈ (32 + 117) - 148.68 ≈ 149 - 148.68 ≈ 0.32 )So, ( f(1.26) ≈ 0.32 )Close to zero. Let's try ( r = 1.258 ):Calculate ( 1.258^{15} ). Hmm, this is getting complicated.Alternatively, let me use linear approximation between ( r = 1.25 ) and ( r = 1.26 ).At ( r = 1.25 ), ( f(r) ≈ -2.0781 )At ( r = 1.26 ), ( f(r) ≈ 0.32 )The change in ( r ) is 0.01, and the change in ( f(r) ) is 0.32 - (-2.0781) = 2.3981We need to find ( r ) such that ( f(r) = 0 ). The distance from ( r = 1.25 ) is ( 2.0781 / 2.3981 ≈ 0.866 ) of the interval.So, ( r ≈ 1.25 + 0.866 * 0.01 ≈ 1.25866 )So, approximately 1.2587.Let me check ( f(1.2587) ):First, calculate ( 1.2587^{15} ). This is time-consuming, but let me approximate.Using logarithms:( ln(1.2587) ≈ 0.231 )So, ( ln(1.2587^{15}) = 15 * 0.231 ≈ 3.465 )So, ( 1.2587^{15} ≈ e^{3.465} ≈ 32 ) (since ( e^{3.465} ≈ 32 ))Then, ( f(1.2587) = 32 - 118*1.2587 + 117 ≈ 32 - 148.68 + 117 ≈ 0.32 )Wait, that's the same as at ( r = 1.26 ). Hmm, maybe my approximation is off.Alternatively, perhaps I need to use a better method.Let me use the Newton-Raphson method.The function is ( f(r) = r^{15} - 118r + 117 )The derivative is ( f'(r) = 15r^{14} - 118 )Starting with an initial guess ( r_0 = 1.26 ), where ( f(r_0) ≈ 0.32 )Compute ( f'(1.26) = 15*(1.26)^{14} - 118 )First, calculate ( 1.26^{14} ):From earlier, ( 1.26^{15} ≈ 32 ), so ( 1.26^{14} ≈ 32 / 1.26 ≈ 25.4 )So, ( f'(1.26) ≈ 15*25.4 - 118 ≈ 381 - 118 ≈ 263 )Then, the next approximation is:( r_1 = r_0 - f(r_0)/f'(r_0) ≈ 1.26 - 0.32/263 ≈ 1.26 - 0.001216 ≈ 1.258784 )So, ( r_1 ≈ 1.258784 )Now, compute ( f(r_1) ):( f(1.258784) = (1.258784)^{15} - 118*(1.258784) + 117 )Again, using logarithms:( ln(1.258784) ≈ 0.231 )So, ( ln(1.258784^{15}) ≈ 15*0.231 ≈ 3.465 )So, ( 1.258784^{15} ≈ e^{3.465} ≈ 32 )Then, ( f(r_1) ≈ 32 - 118*1.258784 + 117 ≈ 32 - 148.68 + 117 ≈ 0.32 )Wait, that's the same as before. Hmm, maybe my approximation of ( 1.258784^{15} ≈ 32 ) is too rough.Alternatively, perhaps I need to use a better approximation for ( 1.258784^{15} ).Let me try to compute ( 1.258784^{15} ) more accurately.Using the fact that ( ln(1.258784) ≈ 0.231 ), so ( ln(1.258784^{15}) ≈ 3.465 ), so ( e^{3.465} ≈ 32 ). But let me check with a calculator:Actually, ( e^{3.465} ≈ e^{3} * e^{0.465} ≈ 20.0855 * 1.592 ≈ 32 ). So, that's correct.So, ( f(r_1) ≈ 32 - 148.68 + 117 ≈ 0.32 )Wait, that's the same as before. So, maybe my method isn't converging quickly enough.Alternatively, perhaps I need to accept that ( r ≈ 1.2587 ) is a good enough approximation.So, rounding to four decimal places, ( r ≈ 1.2587 )But let me check with ( r = 1.2587 ):Calculate ( 1.2587^{15} ) more accurately.Using a calculator:1.2587^1 = 1.25871.2587^2 ≈ 1.2587 * 1.2587 ≈ 1.58411.2587^3 ≈ 1.5841 * 1.2587 ≈ 1.9931.2587^4 ≈ 1.993 * 1.2587 ≈ 2.5071.2587^5 ≈ 2.507 * 1.2587 ≈ 3.1561.2587^6 ≈ 3.156 * 1.2587 ≈ 3.9721.2587^7 ≈ 3.972 * 1.2587 ≈ 5.0001.2587^8 ≈ 5.000 * 1.2587 ≈ 6.29351.2587^9 ≈ 6.2935 * 1.2587 ≈ 7.9161.2587^10 ≈ 7.916 * 1.2587 ≈ 9.9681.2587^11 ≈ 9.968 * 1.2587 ≈ 12.551.2587^12 ≈ 12.55 * 1.2587 ≈ 15.791.2587^13 ≈ 15.79 * 1.2587 ≈ 19.851.2587^14 ≈ 19.85 * 1.2587 ≈ 25.001.2587^15 ≈ 25.00 * 1.2587 ≈ 31.4675So, ( r^{15} ≈ 31.4675 )Then, ( f(r) = 31.4675 - 118*1.2587 + 117 ≈ 31.4675 - 148.68 + 117 ≈ (31.4675 + 117) - 148.68 ≈ 148.4675 - 148.68 ≈ -0.2125 )So, ( f(1.2587) ≈ -0.2125 )So, now, ( f(r) ) is negative at ( r = 1.2587 ), and positive at ( r = 1.26 ). So, the root is between 1.2587 and 1.26.Let me try ( r = 1.259 ):Calculate ( 1.259^{15} ). Using the same method:1.259^1 = 1.2591.259^2 ≈ 1.259 * 1.259 ≈ 1.5851.259^3 ≈ 1.585 * 1.259 ≈ 1.9951.259^4 ≈ 1.995 * 1.259 ≈ 2.5091.259^5 ≈ 2.509 * 1.259 ≈ 3.1611.259^6 ≈ 3.161 * 1.259 ≈ 3.9761.259^7 ≈ 3.976 * 1.259 ≈ 5.0081.259^8 ≈ 5.008 * 1.259 ≈ 6.3031.259^9 ≈ 6.303 * 1.259 ≈ 7.9271.259^10 ≈ 7.927 * 1.259 ≈ 10.0001.259^11 ≈ 10.000 * 1.259 ≈ 12.591.259^12 ≈ 12.59 * 1.259 ≈ 15.831.259^13 ≈ 15.83 * 1.259 ≈ 19.911.259^14 ≈ 19.91 * 1.259 ≈ 25.071.259^15 ≈ 25.07 * 1.259 ≈ 31.57So, ( r^{15} ≈ 31.57 )Then, ( f(r) = 31.57 - 118*1.259 + 117 ≈ 31.57 - 148.662 + 117 ≈ (31.57 + 117) - 148.662 ≈ 148.57 - 148.662 ≈ -0.092 )Still negative. Let me try ( r = 1.2595 ):1.2595^15 ≈ ?Using the same method, but this is getting too time-consuming. Alternatively, let me use linear approximation between ( r = 1.2587 ) and ( r = 1.26 ).At ( r = 1.2587 ), ( f(r) ≈ -0.2125 )At ( r = 1.26 ), ( f(r) ≈ 0.32 )We need ( f(r) = 0 ). The change in ( r ) is 0.0013, and the change in ( f(r) ) is 0.32 - (-0.2125) = 0.5325So, the fraction needed is 0.2125 / 0.5325 ≈ 0.4So, ( r ≈ 1.2587 + 0.4 * 0.0013 ≈ 1.2587 + 0.00052 ≈ 1.25922 )So, approximately 1.2592.Let me check ( r = 1.2592 ):Calculate ( 1.2592^{15} ). Again, this is tedious, but let's approximate.Using logarithms:( ln(1.2592) ≈ 0.231 )So, ( ln(1.2592^{15}) ≈ 15*0.231 ≈ 3.465 )So, ( 1.2592^{15} ≈ e^{3.465} ≈ 32 )Then, ( f(r) = 32 - 118*1.2592 + 117 ≈ 32 - 148.68 + 117 ≈ 0.32 )Wait, that's the same as before. Hmm, maybe my approximation is off.Alternatively, perhaps I need to accept that ( r ≈ 1.259 ) is a good enough approximation.So, rounding to four decimal places, ( r ≈ 1.259 )But let me check with ( r = 1.259 ):As before, ( f(r) ≈ -0.092 )Wait, that's still negative. So, maybe I need to go a bit higher.Let me try ( r = 1.2595 ):Using the same method, ( f(r) ≈ 0 ) somewhere around here.Alternatively, perhaps I can accept that ( r ≈ 1.259 ) is close enough.So, after all this, I think the common ratio ( r ) is approximately 1.259.But let me check if this makes sense.If ( r ≈ 1.259 ), then the distances between forts are increasing by about 25.9% each time.The first interval is ( a ≈ 118 / frac{r^{15} - 1}{r - 1} ≈ 118 / 119.23 ≈ 0.989 ) km, as calculated earlier.So, the first interval is about 0.989 km, then 0.989*1.259 ≈ 1.246 km, then 1.246*1.259 ≈ 1.568 km, and so on, up to the 15th interval.The total sum should be approximately 118 km.So, I think this is a reasonable approximation.Therefore, the common ratio is approximately 1.259.But let me check if this is the correct approach.Wait, the problem didn't specify that the first interval is 1 km, so I assumed ( a = 1 ) for simplicity. But in reality, ( a ) can be any value, and the ratio ( r ) would adjust accordingly.But without knowing ( a ), I can't find a unique ( r ). So, perhaps the problem expects me to assume ( a = 1 ), leading to ( r ≈ 1.259 ).Alternatively, maybe the problem is designed so that the ratio is such that the sum is 118, regardless of ( a ). But that's not possible because ( a ) affects the sum.Wait, maybe the problem is designed so that the ratio is such that the sum is 118, and the first term is 1. So, ( r ≈ 1.259 ).Alternatively, perhaps the problem expects a simpler answer, like ( r = sqrt[15]{(118(r - 1) + 1)} ), but that's not helpful.Wait, maybe I can express ( r ) in terms of the total sum.But I think I've spent enough time on this, and the best approximation I can get is ( r ≈ 1.259 ).So, for the first problem, the common ratio is approximately 1.259.Now, moving on to the second problem:The total length of Roman roads connected to Hadrian's Wall is modeled by the sum of an infinite geometric series. The first road segment is 10 km, and the common ratio is 0.8. Calculate the total length.Okay, this seems straightforward.The sum of an infinite geometric series is given by ( S = frac{a}{1 - r} ), where ( a ) is the first term and ( r ) is the common ratio, provided that ( |r| < 1 ).Here, ( a = 10 ) km, ( r = 0.8 ).So, ( S = frac{10}{1 - 0.8} = frac{10}{0.2} = 50 ) km.So, the total length is 50 km.That seems straightforward.But let me double-check.Yes, the formula for the sum of an infinite geometric series is ( S = frac{a}{1 - r} ) when ( |r| < 1 ). Here, ( r = 0.8 ), which is less than 1, so the formula applies.So, ( S = 10 / (1 - 0.8) = 10 / 0.2 = 50 ) km.Yes, that's correct.So, the total length of the Roman roads connected to Hadrian's Wall is 50 kilometers.Therefore, the answers are:1. The common ratio is approximately 1.259.2. The total length of the roads is 50 km.But let me check if the first problem can be solved more accurately.Wait, I think I made a mistake in assuming ( a = 1 ). The problem doesn't specify ( a ), so perhaps the ratio can be found without knowing ( a ).Wait, the sum is 118 km, which is the sum of 15 terms of a geometric series with first term ( a ) and ratio ( r ).So, ( S = a times frac{r^{15} - 1}{r - 1} = 118 )But without knowing ( a ), I can't solve for ( r ). So, perhaps the problem is designed so that ( a = 1 ), leading to ( r ≈ 1.259 ).Alternatively, maybe the problem expects me to express ( r ) in terms of the total sum, but that's not possible without knowing ( a ).Wait, perhaps the problem is designed so that the ratio is such that the sum is 118, and the first term is 1, leading to ( r ≈ 1.259 ).Alternatively, maybe the problem expects me to recognize that the sum is 118, and the number of terms is 15, so the ratio can be found using the formula.But without knowing ( a ), I can't find ( r ). So, perhaps the problem expects me to assume ( a = 1 ), leading to ( r ≈ 1.259 ).Alternatively, maybe the problem is designed so that the ratio is such that the sum is 118, and the first term is 1, leading to ( r ≈ 1.259 ).So, I think that's the best I can do for the first problem.Therefore, the answers are:1. The common ratio is approximately 1.259.2. The total length of the roads is 50 km.</think>

🧠:<think>Okay, so I have two calculus problems to solve related to an artist's work. Let me take them one by one.Starting with the first problem: The artist is creating symmetrical designs by folding the paper along its diagonal and applying paint. The paint spreads according to the function ( f(x, y) = e^{-(x^2 + y^2)} ). I need to calculate the total amount of paint applied to one half of the canvas, specifically the half-plane where ( y geq x ).Hmm, so the canvas is a square with side length 1 meter. Folding along the diagonal would mean we're dealing with symmetry across the line ( y = x ). The half-plane ( y geq x ) is one of the two regions created by this fold. Since the paint spreads according to ( f(x, y) ), the total paint applied to this half-plane would be the double integral of ( f(x, y) ) over the region where ( y geq x ).But wait, the canvas is a square, so the coordinates go from (0,0) to (1,1). So, the region ( y geq x ) within the square is the area above the diagonal from (0,0) to (1,1). That makes sense.So, the integral I need to compute is:[iint_{D} e^{-(x^2 + y^2)} , dA]where ( D ) is the region ( y geq x ) within the unit square.Since the region is a triangle, maybe it's easier to switch to polar coordinates? But wait, the square might complicate things because polar coordinates are more suited for circular regions. Alternatively, maybe I can set up the integral in Cartesian coordinates.Let me sketch the region. It's the area above the line ( y = x ) within the square [0,1] x [0,1]. So, for each x from 0 to 1, y goes from x to 1.So, the integral becomes:[int_{0}^{1} int_{x}^{1} e^{-(x^2 + y^2)} , dy , dx]Hmm, integrating this directly might be tricky because the integrand is ( e^{-(x^2 + y^2)} ), which doesn't separate nicely into x and y parts. Maybe switching to polar coordinates would help, but since the region isn't a full circle, it's a bit more complicated.Wait, but the exponent is ( -(x^2 + y^2) ), which is similar to the Gaussian function. The integral over the entire plane is known, but here we're only integrating over a specific region.Alternatively, maybe I can exploit symmetry. Since the function ( f(x, y) ) is symmetric with respect to the line ( y = x ), the integral over ( y geq x ) should be equal to the integral over ( y leq x ). Therefore, the total integral over the entire square would be twice the integral over ( y geq x ).But wait, the entire square's integral is:[int_{0}^{1} int_{0}^{1} e^{-(x^2 + y^2)} , dy , dx]Which is equal to ( left( int_{0}^{1} e^{-x^2} , dx right)^2 ), since the integrand factors into ( e^{-x^2} e^{-y^2} ).So, if I compute this, then the integral over ( y geq x ) would be half of that.But wait, is that correct? Because the function is symmetric, yes, the integral over each half should be equal. So, the total integral is ( left( int_{0}^{1} e^{-x^2} dx right)^2 ), which is approximately known, but I need to compute it exactly or express it in terms of known functions.Wait, the integral ( int e^{-x^2} dx ) is related to the error function, erf(x). Specifically,[int_{0}^{1} e^{-x^2} dx = frac{sqrt{pi}}{2} text{erf}(1)]So, the total integral over the square is ( left( frac{sqrt{pi}}{2} text{erf}(1) right)^2 ). Therefore, the integral over ( y geq x ) is half of that, which is:[frac{pi}{4} left( text{erf}(1) right)^2]But wait, is that correct? Let me think again. The function ( e^{-(x^2 + y^2)} ) is symmetric in x and y, so integrating over the entire square [0,1]x[0,1] would indeed be the square of the integral from 0 to 1 of ( e^{-x^2} dx ). Therefore, the integral over the region ( y geq x ) is half of that.So, the total paint applied to one half is:[frac{1}{2} left( int_{0}^{1} e^{-x^2} dx right)^2 = frac{1}{2} left( frac{sqrt{pi}}{2} text{erf}(1) right)^2 = frac{pi}{8} left( text{erf}(1) right)^2]But I should check if this is correct. Alternatively, maybe I can compute the integral in polar coordinates.In polar coordinates, the region ( y geq x ) corresponds to angles from ( pi/4 ) to ( 5pi/4 ), but since we're in the unit square, the limits are more complicated because the square doesn't extend beyond r = 1 in all directions. So, polar coordinates might not be straightforward here.Alternatively, maybe I can switch the order of integration. Let me see:Original integral:[int_{0}^{1} int_{x}^{1} e^{-(x^2 + y^2)} dy dx]If I switch the order, for y from 0 to 1, x goes from 0 to y.So, it becomes:[int_{0}^{1} int_{0}^{y} e^{-(x^2 + y^2)} dx dy]But that doesn't seem to help much because the integrand is still ( e^{-(x^2 + y^2)} ), which is difficult to integrate with respect to x or y.Alternatively, maybe I can use a substitution. Let me set u = x^2 + y^2. But then, the differential would involve both x and y, which complicates things.Wait, perhaps I can use a substitution where I let u = x and v = y - x or something like that. Let me try a change of variables.Let me set u = x, v = y - x. Then, the Jacobian determinant is:[frac{partial(u, v)}{partial(x, y)} = begin{vmatrix} 1 & 0 -1 & 1 end{vmatrix} = 1]So, the area element remains the same. The region ( y geq x ) becomes ( v geq 0 ). But the limits for u and v would still be within the square, which complicates things because u ranges from 0 to 1, and v ranges from -u to 1 - u, but since we're considering v >= 0, it's from 0 to 1 - u.So, the integral becomes:[int_{u=0}^{1} int_{v=0}^{1 - u} e^{-(u^2 + (u + v)^2)} dv du]Hmm, that seems more complicated. Maybe not helpful.Alternatively, perhaps I can use a substitution where I set x = r cosθ, y = r sinθ, but again, the square complicates the limits.Wait, maybe I can extend the region to the entire first quadrant and then subtract the parts outside the square, but that might not be straightforward either.Alternatively, perhaps I can use a Monte Carlo method, but since this is a theoretical problem, I need an exact answer.Wait, maybe I can express the integral as a product of two integrals if I can separate variables, but the exponent is ( x^2 + y^2 ), which doesn't separate, so that won't work.Alternatively, maybe I can expand ( e^{-(x^2 + y^2)} ) as a power series and integrate term by term.Recall that ( e^{-z} = sum_{n=0}^{infty} frac{(-z)^n}{n!} ). So,[e^{-(x^2 + y^2)} = sum_{n=0}^{infty} frac{(-1)^n (x^2 + y^2)^n}{n!}]Then, the integral becomes:[sum_{n=0}^{infty} frac{(-1)^n}{n!} int_{0}^{1} int_{x}^{1} (x^2 + y^2)^n dy dx]But integrating ( (x^2 + y^2)^n ) over the region is still complicated. Maybe using binomial expansion:[(x^2 + y^2)^n = sum_{k=0}^{n} binom{n}{k} x^{2k} y^{2(n - k)}]So, the integral becomes:[sum_{n=0}^{infty} frac{(-1)^n}{n!} sum_{k=0}^{n} binom{n}{k} int_{0}^{1} x^{2k} int_{x}^{1} y^{2(n - k)} dy dx]This seems very involved, but perhaps manageable.First, compute the inner integral over y:[int_{x}^{1} y^{2(n - k)} dy = left[ frac{y^{2(n - k) + 1}}{2(n - k) + 1} right]_{x}^{1} = frac{1}{2(n - k) + 1} - frac{x^{2(n - k) + 1}}{2(n - k) + 1}]So, the integral becomes:[sum_{n=0}^{infty} frac{(-1)^n}{n!} sum_{k=0}^{n} binom{n}{k} left( frac{1}{2(n - k) + 1} - frac{1}{2(n - k) + 1} int_{0}^{1} x^{2(n - k) + 1 + 2k} dx right)]Simplify the exponent in the x integral:( 2(n - k) + 1 + 2k = 2n + 1 )So, the integral over x is:[int_{0}^{1} x^{2n + 1} dx = frac{1}{2n + 2}]Therefore, the entire expression becomes:[sum_{n=0}^{infty} frac{(-1)^n}{n!} sum_{k=0}^{n} binom{n}{k} left( frac{1}{2(n - k) + 1} - frac{1}{(2(n - k) + 1)(2n + 2)} right)]This is getting really complicated. Maybe there's a better approach.Wait, going back, perhaps I can use the fact that the integral over the entire square is ( left( int_{0}^{1} e^{-x^2} dx right)^2 ), and since the function is symmetric, the integral over ( y geq x ) is half of that. So, maybe I can just compute ( frac{1}{2} left( int_{0}^{1} e^{-x^2} dx right)^2 ).But is that correct? Let me think. The function ( e^{-(x^2 + y^2)} ) is symmetric in x and y, so yes, the integral over ( y geq x ) should be equal to the integral over ( y leq x ). Therefore, the total integral over the square is twice the integral over ( y geq x ). So, the integral over ( y geq x ) is half of the total integral.Therefore, the total amount of paint is:[frac{1}{2} left( int_{0}^{1} e^{-x^2} dx right)^2]And since ( int_{0}^{1} e^{-x^2} dx = frac{sqrt{pi}}{2} text{erf}(1) ), then:[frac{1}{2} left( frac{sqrt{pi}}{2} text{erf}(1) right)^2 = frac{pi}{8} left( text{erf}(1) right)^2]So, that's the answer for the first problem.Now, moving on to the second problem: The artist creates a pattern using a combination of sinusoidal curves and radial symmetry, described by the polar equation ( r(theta) = sin(4theta) ). I need to determine the area of the region enclosed by this curve on the canvas.First, I recall that in polar coordinates, the area enclosed by a curve ( r = f(theta) ) is given by:[A = frac{1}{2} int_{alpha}^{beta} [f(theta)]^2 dtheta]where ( alpha ) and ( beta ) are the angles where the curve starts and ends. For ( r = sin(4theta) ), this is a polar curve known as a rose curve. The number of petals depends on the coefficient of ( theta ). Since 4 is even, the rose has 8 petals.But wait, actually, for ( r = sin(ntheta) ), if n is even, it has 2n petals, and if n is odd, it has n petals. So, for n=4, it has 8 petals.However, the artist is working on a square canvas of side length 1 meter. So, the entire curve might not fit within the canvas, but I think the problem assumes that the curve is drawn on the canvas, so we need to compute the area enclosed by the curve within the canvas.But wait, the canvas is a square, but the curve is in polar coordinates. So, perhaps the entire curve is within the unit circle, which is inscribed within the square. So, the maximum radius is 1, so the curve ( r = sin(4theta) ) will oscillate between 0 and 1.But let me confirm: ( sin(4theta) ) ranges between -1 and 1, but since r cannot be negative in polar coordinates (or can it?), depending on the convention. If r can be negative, then the curve would have petals in both positive and negative directions, but since the canvas is a square, negative r would correspond to points in the opposite direction.But in any case, the area enclosed by the entire rose curve ( r = sin(4theta) ) is given by:[A = frac{1}{2} int_{0}^{2pi} [sin(4theta)]^2 dtheta]But since the curve is symmetric, we can compute the area for one petal and multiply by the number of petals.Wait, for ( r = sin(4theta) ), each petal is traced out as ( theta ) goes from 0 to ( pi/2 ). Because the period of ( sin(4theta) ) is ( pi/2 ). So, over ( 0 ) to ( 2pi ), it completes 8 petals.But actually, each petal is formed over an interval of ( pi/4 ), because the rose with 8 petals has each petal spanning ( pi/4 ) radians.Wait, let me think again. For ( r = sin(ntheta) ), the number of petals is n if n is odd, and 2n if n is even. So, for n=4, it's 8 petals. Each petal is formed over an interval of ( pi/4 ) because the period is ( pi/2 ), and each petal is traced twice in that period. Wait, no, actually, each petal is formed over ( pi/2 ) divided by 2, which is ( pi/4 ).Wait, perhaps it's better to compute the area over one petal and multiply by 8.But let's compute the integral over the full period.The area is:[A = frac{1}{2} int_{0}^{2pi} sin^2(4theta) dtheta]Using the identity ( sin^2(x) = frac{1 - cos(2x)}{2} ), we can rewrite the integral as:[A = frac{1}{2} int_{0}^{2pi} frac{1 - cos(8theta)}{2} dtheta = frac{1}{4} int_{0}^{2pi} [1 - cos(8theta)] dtheta]Integrating term by term:[frac{1}{4} left[ int_{0}^{2pi} 1 dtheta - int_{0}^{2pi} cos(8theta) dtheta right]]Compute the integrals:First integral: ( int_{0}^{2pi} 1 dtheta = 2pi )Second integral: ( int_{0}^{2pi} cos(8theta) dtheta ). The integral of ( cos(ktheta) ) over ( 0 ) to ( 2pi ) is zero for any integer k ≠ 0. So, this integral is zero.Therefore, the area is:[frac{1}{4} times 2pi = frac{pi}{2}]But wait, that's the area of the entire rose curve. However, the artist's canvas is a square of side length 1, so the maximum radius is 1. But the rose curve ( r = sin(4theta) ) has a maximum radius of 1, so it fits within the unit circle, which is inscribed in the square. Therefore, the area enclosed by the curve on the canvas is ( frac{pi}{2} ).But wait, is that correct? Because the rose curve ( r = sin(4theta) ) actually has 8 petals, each with area ( frac{pi}{16} ), so total area ( 8 times frac{pi}{16} = frac{pi}{2} ). Yes, that matches.So, the area enclosed by the curve on the canvas is ( frac{pi}{2} ) square meters.But wait, let me double-check. The integral over ( 0 ) to ( 2pi ) gives the total area of the rose curve, which is ( frac{pi}{2} ). Since the canvas is a square of side 1, the unit circle is inscribed within it, so the entire rose curve lies within the canvas. Therefore, the area is indeed ( frac{pi}{2} ).Alternatively, if the artist is drawing the curve on the square canvas, but the curve might extend beyond the square? Wait, no, because the maximum radius is 1, and the square has side length 1, so the circle of radius 1 is inscribed in the square. Therefore, the rose curve, which is within the unit circle, is entirely within the square. So, the area is ( frac{pi}{2} ).So, to summarize:1. The total paint applied to one half of the canvas is ( frac{pi}{8} left( text{erf}(1) right)^2 ).2. The area enclosed by the polar curve ( r(theta) = sin(4theta) ) is ( frac{pi}{2} ).But wait, for the first problem, I think I made a mistake. The integral over the entire square is ( left( int_{0}^{1} e^{-x^2} dx right)^2 ), and since the function is symmetric, the integral over ( y geq x ) is half of that. However, I should confirm if the function is symmetric in such a way that the integral over ( y geq x ) is exactly half.Yes, because for every point (x, y) in ( y geq x ), there is a corresponding point (y, x) in ( y leq x ), and the function ( f(x, y) = e^{-(x^2 + y^2)} ) is symmetric in x and y. Therefore, the integrals over both regions are equal, so each is half of the total integral over the square.Therefore, the first answer is correct.So, final answers:1. ( frac{pi}{8} left( text{erf}(1) right)^2 )2. ( frac{pi}{2} )But wait, erf(1) is a specific value. Maybe I can express it in terms of erf(1), or perhaps the problem expects a numerical value? The problem says to use advanced calculus and integration techniques, so expressing it in terms of erf is acceptable.Alternatively, maybe I can write it as ( frac{pi}{8} text{erf}(1)^2 ).Yes, that's fine.</think>