🧠:<think>Alright, so I have two problems here related to energy consumption and renewable energy targets for a factory. Let me take them one by one.Starting with the first problem: The factory's total energy consumption is modeled by the function ( E(t) = 5000 + 300sin(2pi t) + 200cos(2pi t) ) MWh, where ( t ) is the time in years. I need to calculate the average energy consumption per day over the year.Hmm, okay. So, average energy consumption over a year. Since the function is given in terms of a year, I think I need to find the average value of ( E(t) ) over one year. The average value of a function over an interval can be found by integrating the function over that interval and then dividing by the length of the interval.The function is periodic with period 1 year because the sine and cosine functions have arguments ( 2pi t ), which means their periods are 1 year. So, to find the average, I can integrate ( E(t) ) from 0 to 1 and then divide by 1, which is just the integral from 0 to 1.Let me write that down:Average energy consumption per year, ( overline{E} = frac{1}{1-0} int_{0}^{1} E(t) dt = int_{0}^{1} [5000 + 300sin(2pi t) + 200cos(2pi t)] dt ).Now, integrating term by term:1. The integral of 5000 with respect to t is 5000t.2. The integral of ( 300sin(2pi t) ) with respect to t is ( -frac{300}{2pi} cos(2pi t) ).3. The integral of ( 200cos(2pi t) ) with respect to t is ( frac{200}{2pi} sin(2pi t) ).So putting it all together:( overline{E} = [5000t - frac{300}{2pi} cos(2pi t) + frac{200}{2pi} sin(2pi t)] ) evaluated from 0 to 1.Now, evaluating at t = 1:- ( 5000(1) = 5000 )- ( cos(2pi * 1) = cos(2pi) = 1 )- ( sin(2pi * 1) = sin(2pi) = 0 )So, at t = 1: ( 5000 - frac{300}{2pi}(1) + frac{200}{2pi}(0) = 5000 - frac{300}{2pi} ).Evaluating at t = 0:- ( 5000(0) = 0 )- ( cos(0) = 1 )- ( sin(0) = 0 )So, at t = 0: ( 0 - frac{300}{2pi}(1) + frac{200}{2pi}(0) = - frac{300}{2pi} ).Subtracting the lower limit from the upper limit:( [5000 - frac{300}{2pi}] - [- frac{300}{2pi}] = 5000 - frac{300}{2pi} + frac{300}{2pi} = 5000 ).So, the average energy consumption per year is 5000 MWh.But wait, the question asks for the average energy consumption per day over the year. So, I need to convert this annual average into a daily average.There are 365 days in a year, so average per day would be ( frac{5000}{365} ) MWh per day.Let me compute that:5000 divided by 365. Let me see, 365 times 13 is 4745, and 365 times 14 is 5110. So, 5000 is between 13 and 14 times 365.Compute 5000 / 365:Divide numerator and denominator by 5: 1000 / 73 ≈ 13.6986.So approximately 13.6986 MWh per day.But let me do it more accurately:365 * 13 = 47455000 - 4745 = 255So, 255 / 365 = 0.6986So, total is 13.6986 MWh per day.So, approximately 13.7 MWh per day.Wait, but let me check if I did everything correctly.First, the average over the year is 5000 MWh/year. Then, dividing by 365 days gives approximately 13.7 MWh/day.Yes, that seems correct.Alternatively, since the sine and cosine functions have an average of zero over a full period, the average consumption is just the constant term, which is 5000 MWh/year. So, that's another way to see it without integrating.So, 5000 MWh/year divided by 365 days is indeed approximately 13.7 MWh/day.So, the average energy consumption per day over the year is approximately 13.7 MWh.Moving on to the second problem: The factory aims to increase its renewable energy usage to 70% of its total energy consumption by installing additional solar panels. Currently, the factory generates renewable energy modeled by ( R(t) = 1500 + 100sin(2pi t) + 50cos(2pi t) ) MWh per year. Determine the additional capacity ( C ) (in MWh) of renewable energy needed per year to achieve the 70% target.Alright, so first, let's understand what is being asked. The factory wants 70% of its total energy consumption to come from renewable sources. Currently, it's using ( R(t) ) MWh of renewable energy per year, and the total energy consumption is ( E(t) ) MWh per year.So, the target is that renewable energy should be 70% of total consumption. So, ( R(t) + C = 0.7 times E(t) ), where ( C ) is the additional capacity needed.But wait, actually, both ( R(t) ) and ( E(t) ) are functions of time. However, the problem says "to achieve the 70% target," which I think refers to the total annual consumption. So, perhaps we need to compute the average total consumption and the average renewable consumption, then find the additional capacity needed so that the total renewable is 70% of total consumption.Alternatively, maybe it's considering the functions over the year and integrating them. Let me think.Wait, the first problem gave us the average energy consumption, which was 5000 MWh/year. So, perhaps the total consumption is 5000 MWh/year on average. Similarly, the renewable energy is given by ( R(t) = 1500 + 100sin(2pi t) + 50cos(2pi t) ). So, to find the average renewable energy, we can compute the average of ( R(t) ) over a year.Again, since the sine and cosine terms average out to zero over a full period, the average renewable energy is just 1500 MWh/year.So, currently, the factory uses 1500 MWh/year of renewable energy and 5000 MWh/year total. So, the current percentage is ( frac{1500}{5000} = 0.3 ) or 30%.They want to increase this to 70%, so the required renewable energy is 0.7 * 5000 = 3500 MWh/year.Therefore, the additional capacity needed is 3500 - 1500 = 2000 MWh/year.Wait, but hold on. Is it that straightforward? Because both ( E(t) ) and ( R(t) ) are time-dependent functions. So, maybe we need to compute the total energy over the year and the total renewable energy over the year, then find the required additional capacity.But since both functions are periodic with period 1 year, their integrals over a year will give the total consumption and total renewable energy.So, let me compute the total energy consumption over a year:( int_{0}^{1} E(t) dt = int_{0}^{1} [5000 + 300sin(2pi t) + 200cos(2pi t)] dt ).We already did this integral earlier and found it to be 5000 MWh/year.Similarly, the total renewable energy over a year is ( int_{0}^{1} R(t) dt = int_{0}^{1} [1500 + 100sin(2pi t) + 50cos(2pi t)] dt ).Again, integrating term by term:1. Integral of 1500 is 1500t.2. Integral of ( 100sin(2pi t) ) is ( -frac{100}{2pi} cos(2pi t) ).3. Integral of ( 50cos(2pi t) ) is ( frac{50}{2pi} sin(2pi t) ).Evaluating from 0 to 1:At t=1:- 1500(1) = 1500- ( -frac{100}{2pi} cos(2pi) = -frac{100}{2pi} (1) = -frac{100}{2pi} )- ( frac{50}{2pi} sin(2pi) = 0 )At t=0:- 1500(0) = 0- ( -frac{100}{2pi} cos(0) = -frac{100}{2pi} (1) = -frac{100}{2pi} )- ( frac{50}{2pi} sin(0) = 0 )Subtracting lower limit from upper limit:( [1500 - frac{100}{2pi}] - [0 - frac{100}{2pi}] = 1500 - frac{100}{2pi} + frac{100}{2pi} = 1500 ).So, the total renewable energy over a year is 1500 MWh.Therefore, the total energy consumption is 5000 MWh/year, and the total renewable energy is 1500 MWh/year. So, the current percentage is 1500 / 5000 = 0.3, or 30%.They want to increase this to 70%, so the required renewable energy is 0.7 * 5000 = 3500 MWh/year.Thus, the additional capacity needed is 3500 - 1500 = 2000 MWh/year.Wait, but let me think again. Is the renewable energy function ( R(t) ) already including some variability, but the total over the year is 1500 MWh. So, if we add a constant capacity ( C ), the total renewable energy becomes ( 1500 + C ). So, we need ( 1500 + C = 0.7 * 5000 ), so ( C = 3500 - 1500 = 2000 ).Yes, that seems correct.Alternatively, if we were to model the additional capacity as a constant function, then integrating ( R(t) + C ) over the year would give ( 1500 + C ). So, setting ( 1500 + C = 0.7 * 5000 ), which gives ( C = 2000 ).Therefore, the additional capacity needed is 2000 MWh/year.But wait, let me check if the renewable energy is supposed to be 70% at all times or on average. The problem says "to achieve the 70% target," which I think refers to the total annual consumption. So, on average, 70% of the total energy comes from renewables. So, the total renewable energy should be 70% of total consumption.Therefore, yes, 0.7 * 5000 = 3500, so additional capacity is 2000.So, summarizing:1. The average energy consumption per day is approximately 13.7 MWh.2. The additional renewable capacity needed is 2000 MWh/year.Final Answer1. The average energy consumption per day over the year is boxed{13.7} MWh.2. The additional capacity ( C ) needed is boxed{2000} MWh per year.</think>

🧠:<think>Alright, so I have this problem about a nature enthusiast exploring the Amazon rainforest. They’re planning a closed loop through three different ecosystems, and the path is represented by a triangle with vertices at points A(2, 3), B(8, 11), and C(-4, 7). I need to calculate two things: the area of the triangle and the total distance traveled, which is the perimeter of the triangle.Starting with the first part, calculating the area. I remember there are a few ways to find the area of a triangle when you have the coordinates of the vertices. One common method is the shoelace formula. I think it involves multiplying coordinates in a specific way and then taking half the absolute difference. Let me recall the formula.The shoelace formula is given by:Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|Alternatively, it can also be written as:Area = (1/2) |(x₁x₂(y₂ - y₃) + x₂x₃(y₃ - y₁) + x₃x₁(y₁ - y₂))|Wait, no, maybe I'm mixing it up. Let me double-check. I think it's more straightforward: you list the coordinates in order, multiply diagonally, subtract, and take half the absolute value.So, writing out the coordinates:A(2, 3), B(8, 11), C(-4, 7)I can set them up in a table for the shoelace formula:x: 2, 8, -4, 2y: 3, 11, 7, 3Then, multiply diagonally down to the right and sum them up:(2*11) + (8*7) + (-4*3) = 22 + 56 - 12 = 66Then, multiply diagonally up to the right and sum them up:(3*8) + (11*(-4)) + (7*2) = 24 - 44 + 14 = 24 - 44 is -20, plus 14 is -6Now, subtract the second sum from the first sum:66 - (-6) = 66 + 6 = 72Take half the absolute value:Area = (1/2)*|72| = 36Wait, that seems straightforward. So the area is 36 square units.But just to make sure, let me try another method, maybe using vectors or the determinant method.Another way is to use the formula:Area = (1/2) | (x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁) |Let me assign A as (x₁, y₁) = (2,3), B as (x₂, y₂) = (8,11), and C as (x₃, y₃) = (-4,7).Compute (x₂ - x₁) = 8 - 2 = 6Compute (y₃ - y₁) = 7 - 3 = 4Compute (x₃ - x₁) = -4 - 2 = -6Compute (y₂ - y₁) = 11 - 3 = 8Now, plug into the formula:Area = (1/2) | (6)(4) - (-6)(8) | = (1/2) |24 - (-48)| = (1/2)|24 + 48| = (1/2)(72) = 36Same result. So that seems consistent. Maybe I can also use vectors to compute the cross product.Vectors AB and AC can be found as:Vector AB = B - A = (8-2, 11-3) = (6,8)Vector AC = C - A = (-4-2, 7-3) = (-6,4)The area is (1/2)|AB × AC|, which is (1/2)|6*4 - 8*(-6)| = (1/2)|24 + 48| = (1/2)(72) = 36Consistent again. So I think 36 is correct.Moving on to the second part: calculating the total distance traveled, which is the perimeter of the triangle. So I need to find the lengths of sides AB, BC, and CA, then sum them up.First, let's find the distance between A and B.Points A(2,3) and B(8,11).Distance formula: sqrt[(x₂ - x₁)² + (y₂ - y₁)²]So, AB = sqrt[(8-2)² + (11-3)²] = sqrt[6² + 8²] = sqrt[36 + 64] = sqrt[100] = 10Okay, that's straightforward.Next, distance between B(8,11) and C(-4,7).BC = sqrt[(-4 - 8)² + (7 - 11)²] = sqrt[(-12)² + (-4)²] = sqrt[144 + 16] = sqrt[160]Simplify sqrt[160]. 160 = 16*10, so sqrt[16*10] = 4*sqrt(10). Approximately, sqrt(10) is about 3.162, so 4*3.162 ≈12.648, but since we need exact value, it's 4√10.Wait, hold on, let me confirm:(-4 - 8) is -12, squared is 144(7 - 11) is -4, squared is 16144 + 16 = 160Yes, sqrt(160) = 4√10. Correct.Now, distance between C(-4,7) and A(2,3).CA = sqrt[(2 - (-4))² + (3 - 7)²] = sqrt[(6)² + (-4)²] = sqrt[36 + 16] = sqrt[52]Simplify sqrt(52). 52 = 4*13, so sqrt(4*13) = 2√13.So, CA is 2√13.Now, let's sum up the sides:AB = 10BC = 4√10CA = 2√13Total distance = 10 + 4√10 + 2√13I think that's the exact value. If needed, we can approximate it numerically, but since the question doesn't specify, exact form is probably better.Let me just verify the calculations once more.AB: between (2,3) and (8,11). Differences: 6 and 8. 6² + 8² = 36 + 64 = 100. sqrt(100)=10. Correct.BC: between (8,11) and (-4,7). Differences: -12 and -4. Squared: 144 and 16. Sum: 160. sqrt(160)=4√10. Correct.CA: between (-4,7) and (2,3). Differences: 6 and -4. Squared: 36 and 16. Sum: 52. sqrt(52)=2√13. Correct.So, adding them up: 10 + 4√10 + 2√13.I don't think these radicals can be combined further, so that's the total distance.Alternatively, if I want to write it as a single expression, it's 10 + 4√10 + 2√13 units.Just to make sure, let me compute approximate values to see if the total makes sense.Compute 4√10: 4*3.162 ≈12.648Compute 2√13: 2*3.606 ≈7.212So total approximate distance: 10 + 12.648 + 7.212 ≈10 + 12.648 is 22.648 +7.212 is 29.86.So approximately 29.86 units. That seems reasonable given the coordinates.Wait, just to check, maybe I can compute the distances again numerically.AB: distance between (2,3) and (8,11):dx=6, dy=8. Distance=10.BC: between (8,11) and (-4,7):dx= -12, dy= -4. Distance sqrt(144 + 16)=sqrt(160)=12.649.CA: between (-4,7) and (2,3):dx=6, dy=-4. Distance sqrt(36 + 16)=sqrt(52)=7.211.So total: 10 + 12.649 + 7.211 ≈29.86.Yes, that's consistent. So the exact value is 10 + 4√10 + 2√13, and approximate is about 29.86.So, summarizing:1. The area of the triangle is 36 square units.2. The total distance traveled is 10 + 4√10 + 2√13 units.I think that's all. I don't see any mistakes in my calculations, and multiple methods for the area gave me the same result, so I'm confident.Final Answer1. The area of the region is boxed{36} square units.2. The total distance traveled is boxed{10 + 4sqrt{10} + 2sqrt{13}} units.</think>

🧠:<think>Okay, so I have this problem about Professor Claire Dubois studying the evolution of dialects in France. She's looking at two dialects, A and B, across 10 regions over the past century. The data is about the percentage of people using specific linguistic features characteristic of each dialect. The problem has two parts. The first part is about expressing the functions f_A(t) and f_B(t) as Fourier series, considering that the frequencies of linguistic features change periodically every 20 years due to sociopolitical events. The second part is about finding the average rate of change of the difference between f_A(t) and f_B(t) over the last century and interpreting it in terms of convergence or divergence of linguistic features between the two dialects.Let me tackle the first part first. Fourier series are used to represent periodic functions as a sum of sine and cosine functions. Since the problem mentions that the frequencies exhibit periodic changes every 20 years, I can assume that the period T is 20 years. The general form of a Fourier series for a function f(t) with period T is:f(t) = a_0 + Σ [a_n cos(2πnt/T) + b_n sin(2πnt/T)]where the sum is from n=1 to infinity, and the coefficients a_n and b_n are calculated using integrals over one period.Given that the interval is from t=0 to t=100, which is 5 periods of 20 years each, the Fourier series should be applicable here.So, for f_A(t) and f_B(t), I can express each as:f_A(t) = a_0 + Σ [a_n cos(πnt/10) + b_n sin(πnt/10)]Similarly,f_B(t) = c_0 + Σ [c_n cos(πnt/10) + d_n sin(πnt/10)]Wait, hold on. The period T is 20 years, so 2π/T would be π/10. Therefore, the arguments of sine and cosine should be (π t)/10. So, the functions can be written as sums of cos(π t /10), cos(2π t /10), etc., and similarly for sine terms.But the problem doesn't specify whether the functions are even or odd, so we need both sine and cosine terms.However, without specific data points or more information about f_A(t) and f_B(t), I can't compute the exact coefficients a_n, b_n, c_n, d_n. The problem just asks to express them as Fourier series, so I think it's sufficient to write the general form.So, for both f_A(t) and f_B(t), the Fourier series would be:f_A(t) = a_0 + Σ [a_n cos(nπ t /10) + b_n sin(nπ t /10)]f_B(t) = c_0 + Σ [c_n cos(nπ t /10) + d_n sin(nπ t /10)]where the summation is from n=1 to infinity.But wait, actually, the standard Fourier series for a function with period T is:f(t) = a_0 + Σ [a_n cos(2πnt/T) + b_n sin(2πnt/T)]So, in this case, T=20, so 2π/T = π/10. Therefore, the terms should be cos(π t /10), cos(2π t /10), etc., which is the same as cos(nπ t /10) for n=1,2,...So, yes, the expressions above are correct.Moving on to the second part. The average rate of change of the linguistic features over the last century is given by:(1/100) ∫₀¹⁰⁰ [d/dt (f_A(t) - f_B(t))] dtI need to compute this integral and interpret it.First, let's simplify the expression inside the integral. The derivative of (f_A(t) - f_B(t)) is f_A’(t) - f_B’(t). Therefore, the integral becomes:(1/100) ∫₀¹⁰⁰ [f_A’(t) - f_B’(t)] dtBut integrating the derivative of a function over an interval gives the difference of the function at the endpoints. So,∫₀¹⁰⁰ f_A’(t) dt = f_A(100) - f_A(0)Similarly,∫₀¹⁰⁰ f_B’(t) dt = f_B(100) - f_B(0)Therefore, the integral becomes:(1/100) [ (f_A(100) - f_A(0)) - (f_B(100) - f_B(0)) ]Simplify this:(1/100) [ (f_A(100) - f_B(100)) - (f_A(0) - f_B(0)) ]So, the average rate of change is:[ (f_A(100) - f_B(100)) - (f_A(0) - f_B(0)) ] / 100This represents the average rate at which the difference between the frequencies of dialect A and B has changed over the century.If this value is positive, it means that the difference (f_A - f_B) has increased on average over the century, implying divergence. If it's negative, the difference has decreased, implying convergence. If it's zero, the average rate of change is zero, meaning no net change in the difference over the century.But without specific values for f_A(100), f_B(100), f_A(0), and f_B(0), I can't compute the exact numerical value. However, the expression itself gives the interpretation.So, summarizing:1. Both f_A(t) and f_B(t) can be expressed as Fourier series with period 20 years, so their expansions will involve sine and cosine terms with frequencies that are integer multiples of π/10.2. The average rate of change of the difference between f_A and f_B is [ (f_A(100) - f_B(100)) - (f_A(0) - f_B(0)) ] / 100. If this is positive, dialects are diverging; if negative, converging; if zero, stable.I think that's the gist of it. I don't have the actual data, so I can't compute numerical coefficients or the exact average rate, but I can set up the expressions as required.Final Answer1. The functions ( f_A(t) ) and ( f_B(t) ) can be expressed using Fourier series with a period of 20 years as: [ f_A(t) = a_0 + sum_{n=1}^{infty} left( a_n cosleft(frac{npi t}{10}right) + b_n sinleft(frac{npi t}{10}right) right) ] [ f_B(t) = c_0 + sum_{n=1}^{infty} left( c_n cosleft(frac{npi t}{10}right) + d_n sinleft(frac{npi t}{10}right) right) ] 2. The average rate of change is (boxed{frac{f_A(100) - f_B(100) - f_A(0) + f_B(0)}{100}}). If positive, dialects diverge; if negative, they converge.</think>

🧠:<think>Okay, so I have this problem where a CEO is evaluating the success of their latest product launch. They have both sales data and customer feedback. The sales data is a time series of monthly sales for the last 12 months, denoted as S = {s₁, s₂, ..., s₁₂}. The customer feedback is also a time series, F = {f₁, f₂, ..., f₁₂}, where each fᵢ is between -1 and 1, representing negative to positive sentiments.There are two sub-problems here. Let me tackle them one by one.Sub-problem 1:They want to define a combined performance metric Pₜ for each month t as Pₜ = αsₜ + βfₜ, where α and β are weights. The constraint is α + β = 1, and the CEO prefers that qualitative feedback (which is β) shouldn't be weighted less than 30%, so β ≥ 0.3.I need to determine the range of possible values for α and β.Hmm, okay. So since α + β = 1, if β is at least 0.3, then α must be at most 0.7. Because if β is 0.3, then α is 0.7. If β increases, α decreases accordingly.So, the range for β is from 0.3 to 1, but wait, since α has to be non-negative as well, right? Because weights can't be negative, I assume. So if β is 1, then α is 0. But the CEO might not want α to be zero because sales are important too. But the problem doesn't specify a lower bound on α, only that β should be at least 0.3.So, the range for β is 0.3 ≤ β ≤ 1, and correspondingly, α is 0 ≤ α ≤ 0.7.Wait, but if β is 1, then α is 0, which might not be desirable, but the problem doesn't say anything about α having a minimum. So, based on the given constraints, β can be as high as 1, making α as low as 0.So, the possible values for α and β are:α ∈ [0, 0.7]β ∈ [0.3, 1]But since α + β = 1, they are dependent on each other. So for each β in [0.3, 1], α is 1 - β, which would be in [0, 0.7].I think that's the answer for Sub-problem 1.Sub-problem 2:Now, the CEO wants to predict future performance using a weighted moving average model based on the past 3 months of combined performance metrics Pₜ. The prediction for month t+1 is given by:hat{P}_{t+1} = (w₁Pₜ + w₂P_{t-1} + w₃P_{t-2}) / (w₁ + w₂ + w₃)But wait, the weights are non-negative and sum to 1, so it's actually:hat{P}_{t+1} = w₁Pₜ + w₂P_{t-1} + w₃P_{t-2}Because if w₁ + w₂ + w₃ = 1, then dividing by 1 doesn't change anything.Given the combined performance data for the last three months: P₁₀ = 80, P₁₁ = 85, P₁₂ = 90.We need to determine the optimal weights w₁, w₂, w₃ that maximize the predicted performance hat{P}_{13}.So, the prediction is a weighted average of the last three months, with weights w₁, w₂, w₃, which are non-negative and sum to 1.To maximize hat{P}_{13}, which is w₁*90 + w₂*85 + w₃*80.Since we want to maximize this, and the weights are non-negative and sum to 1, the maximum occurs when we put as much weight as possible on the highest value, which is 90.So, to maximize, set w₁ = 1, and w₂ = w₃ = 0.Therefore, the optimal weights are w₁ = 1, w₂ = 0, w₃ = 0.But wait, let me think again. Is there any constraint that the weights have to be positive? The problem says non-negative, so zero is allowed.So yes, putting all weight on the most recent month, which has the highest P value, will give the maximum prediction.Alternatively, if the weights had to be positive, we might have to distribute some minimal weight, but since zero is allowed, the optimal is to put all weight on the highest P.So, the optimal weights are w₁ = 1, w₂ = 0, w₃ = 0.I think that's correct.Final AnswerSub-problem 1: The range of possible values is α ∈ [0, 0.7] and β ∈ [0.3, 1]. So, the answer is boxed{alpha in [0, 0.7] text{ and } beta in [0.3, 1]}.Sub-problem 2: The optimal weights are w₁ = 1, w₂ = 0, w₃ = 0. So, the answer is boxed{w_1 = 1, w_2 = 0, w_3 = 0}.</think>