🧠:<think>Okay, so I have this problem about analyzing Nazi propaganda's impact using some statistical models. It's divided into two parts. Let me try to tackle them one by one.Starting with the first part: fitting a logistic growth model to the influence scores of propaganda events. The model is given as ( f(x) = frac{L}{1 + e^{-k(x - x_0)}} ). I need to estimate the parameters L, k, and x0 using the method of least squares. Hmm, least squares usually involves minimizing the sum of squared residuals between the observed data and the model's predictions.First, I should recall what a logistic function looks like. It's an S-shaped curve that starts off slowly, then increases rapidly, and then levels off. The parameters are L, the maximum value; k, the growth rate; and x0, the midpoint where the curve is at half of L.To apply least squares, I need to set up equations that relate the observed influence scores (let's call them y_i) to the model's predictions f(x_i). The residual for each data point is y_i - f(x_i). The least squares method minimizes the sum of the squares of these residuals.So, the objective function to minimize would be:( sum_{i=1}^{n} (y_i - frac{L}{1 + e^{-k(x_i - x_0)}})^2 )To find the estimates of L, k, and x0, I need to take partial derivatives of this objective function with respect to each parameter, set them equal to zero, and solve the resulting system of equations. That sounds like a system of nonlinear equations, which might not have an analytical solution, so I might need to use numerical methods.But the question asks for the equations I would use to derive the parameter estimates. So, I need to write out the partial derivatives.Let me denote the model as:( f(x_i) = frac{L}{1 + e^{-k(x_i - x_0)}} )Then, the partial derivative of the objective function with respect to L is:( frac{partial}{partial L} sum_{i=1}^{n} (y_i - f(x_i))^2 = -2 sum_{i=1}^{n} (y_i - f(x_i)) cdot frac{partial f(x_i)}{partial L} )Similarly, the partial derivatives with respect to k and x0 would involve the derivatives of f with respect to those parameters.Calculating ( frac{partial f}{partial L} ) is straightforward:( frac{partial f}{partial L} = frac{1}{1 + e^{-k(x_i - x_0)}} )For ( frac{partial f}{partial k} ), we have:( frac{partial f}{partial k} = frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot (x_i - x_0) )And for ( frac{partial f}{partial x0} ):( frac{partial f}{partial x0} = frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot k )So, setting each partial derivative equal to zero gives us the normal equations:1. ( sum_{i=1}^{n} (y_i - f(x_i)) cdot frac{1}{1 + e^{-k(x_i - x_0)}} = 0 )2. ( sum_{i=1}^{n} (y_i - f(x_i)) cdot frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot (x_i - x_0) = 0 )3. ( sum_{i=1}^{n} (y_i - f(x_i)) cdot frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot k = 0 )These are the equations I would use to estimate L, k, and x0. However, solving these equations analytically is challenging because they are nonlinear. So, in practice, I would use an iterative numerical method like gradient descent or Newton-Raphson to find the parameter estimates.Moving on to the second part: modeling the number of propaganda events as a Poisson process. I need to calculate the probability that exactly 15 events occurred in a given year, given that the average rate is 12 per year.The Poisson probability mass function is:( P(X = k) = frac{lambda^k e^{-lambda}}{k!} )Where λ is the average rate (12 events per year), and k is the number of occurrences (15 events).So, plugging in the numbers:( P(X = 15) = frac{12^{15} e^{-12}}{15!} )I can compute this value using a calculator or software, but since the question just asks for the probability using the properties of the Poisson distribution, I think expressing it in terms of the formula is sufficient.Wait, but maybe I should compute it approximately? Let me see.Calculating 12^15 is a huge number, and 15! is also large, but the exponential term e^{-12} will dampen it. Alternatively, I can use logarithms to compute it step by step.Alternatively, I can use the natural logarithm to compute the log probability and then exponentiate.But perhaps, for the purposes of this problem, just writing the formula is enough.So, summarizing:1. For the logistic model, set up the partial derivatives of the sum of squared residuals with respect to L, k, and x0, set them to zero, and solve numerically.2. For the Poisson probability, use the PMF formula with λ=12 and k=15.I think that's the gist of it. I don't see any immediate mistakes in my reasoning, but I should double-check the partial derivatives.Wait, when taking the partial derivatives, I should remember that f(x_i) depends on all three parameters, so when taking the derivative with respect to one parameter, the others are treated as constants. So, my expressions for the partial derivatives seem correct.Also, for the Poisson part, I think I got the formula right. The average rate is 12, so λ=12, and we're looking for P(X=15).Yeah, I think that's solid.Final Answer1. The parameter estimates are derived by solving the system of equations obtained from setting the partial derivatives of the sum of squared residuals to zero. The equations are: [ sum_{i=1}^{n} (y_i - frac{L}{1 + e^{-k(x_i - x_0)}}) cdot frac{1}{1 + e^{-k(x_i - x_0)}} = 0 ] [ sum_{i=1}^{n} (y_i - frac{L}{1 + e^{-k(x_i - x_0)}}) cdot frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot (x_i - x_0) = 0 ] [ sum_{i=1}^{n} (y_i - frac{L}{1 + e^{-k(x_i - x_0)}}) cdot frac{L e^{-k(x_i - x_0)}}{(1 + e^{-k(x_i - x_0)})^2} cdot k = 0 ]2. The probability is calculated using the Poisson probability mass function: [ P(X = 15) = frac{12^{15} e^{-12}}{15!} ] Therefore, the final answers are:1. The equations for parameter estimation are as above.2. The probability is boxed{frac{12^{15} e^{-12}}{15!}}.</think>

🧠:<think>Okay, so I have this problem about an influencer promoting a product. Let me try to understand what's being asked here.First, the influencer has 500,000 loyal followers. He's promoting a product that costs R200 each. The conversion rate from followers to buyers is given by this function: C(x) = 1 / (1 + e^{-0.00002(x - 200000)}), where x is the number of followers. So, C(x) is the probability that a follower will buy the product.The first question is asking for the expected number of products sold when the influencer promotes the product to all his followers. The second question is about calculating the expected revenue the influencer makes, considering he gets a 10% commission on each sale.Alright, let's tackle the first part. The expected number of products sold would be the total number of followers multiplied by the conversion rate, right? So, if C(x) is the probability that a single follower buys the product, then for 500,000 followers, the expected sales would be 500,000 * C(500,000).Wait, but hold on. The function C(x) is given in terms of x, which is the number of followers. So, is x the number of followers being targeted, or is it something else? Let me read the problem again."A taxa de conversão de seus seguidores para compradores é modelada pela função C(x) = 1 / (1 + e^{-0,00002(x - 200000)}), onde x é o número de seguidores e C(x) é a probabilidade de conversão de um seguidor em comprador."So, x is the number of followers, and C(x) is the probability that a follower converts into a buyer. So, for each follower, the probability of buying is C(x). Therefore, if the influencer has 500,000 followers, each with a conversion probability of C(500,000), the expected number of sales is 500,000 * C(500,000).Wait, but hold on again. Is x the number of followers being targeted, or is it the total number of followers? Because the function C(x) is defined for x, which is the number of followers. So, if the influencer is promoting to all 500,000 followers, then x is 500,000, and C(500,000) is the conversion rate for each follower.But that seems a bit counterintuitive because usually, conversion rates don't depend on the number of people you're targeting. But in this case, the problem states that the conversion rate is modeled by this function where x is the number of followers. So, perhaps the more followers you have, the higher the conversion rate? Or maybe it's the opposite?Looking at the function: C(x) = 1 / (1 + e^{-0.00002(x - 200000)}). This looks like a logistic function. The standard logistic function is S-shaped, starting near 0, increasing, and approaching 1 as x increases. The parameter here is 0.00002, which is a very small number, so the function will increase very slowly.The midpoint of the logistic function is at x = 200,000, where the exponent becomes zero, so C(200,000) = 1 / (1 + e^0) = 1/2. So, at 200,000 followers, the conversion rate is 50%. For x less than 200,000, the conversion rate is less than 50%, and for x greater than 200,000, it's more than 50%.So, in this case, the influencer has 500,000 followers, which is way more than 200,000. So, the conversion rate should be quite high. Let me calculate C(500,000).First, compute the exponent: -0.00002*(500,000 - 200,000) = -0.00002*(300,000) = -6. So, e^{-6} is approximately... e^6 is about 403.4288, so e^{-6} is about 1/403.4288 ≈ 0.00248.Therefore, C(500,000) = 1 / (1 + 0.00248) ≈ 1 / 1.00248 ≈ 0.99752.So, the conversion rate is approximately 99.752%. That seems extremely high. Is that correct? Let me double-check.Wait, 0.00002 is a very small coefficient, so the exponent is -0.00002*(x - 200,000). So, for x = 500,000, it's -0.00002*300,000 = -6. So, yes, e^{-6} is about 0.00248. So, 1 / (1 + 0.00248) ≈ 0.9975. So, yes, the conversion rate is about 99.75%.That seems extraordinarily high, but maybe in this model, it's the case. So, moving forward with that.Therefore, the expected number of products sold is 500,000 * 0.9975 ≈ 498,750.Wait, but let me compute it more accurately. 0.99752 * 500,000.0.99752 * 500,000 = 500,000 - (500,000 * 0.00248) = 500,000 - 1,240 = 498,760.Wait, because 500,000 * 0.00248 is 1,240. So, 500,000 - 1,240 = 498,760.So, approximately 498,760 products sold.But let me check if I did the exponent correctly. The function is C(x) = 1 / (1 + e^{-0.00002(x - 200000)}). So, for x = 500,000, it's 1 / (1 + e^{-0.00002*(500,000 - 200,000)}) = 1 / (1 + e^{-0.00002*300,000}) = 1 / (1 + e^{-6}).Yes, that's correct. So, e^{-6} is approximately 0.002478752. So, 1 / (1 + 0.002478752) ≈ 0.997521285.So, 500,000 * 0.997521285 ≈ 498,760.6425. So, approximately 498,761 products.But since we can't sell a fraction of a product, we can round it to 498,761.Wait, but maybe the question expects an exact expression rather than an approximate value. Let me see.Alternatively, maybe I can express it in terms of e^{-6}, but I think the question expects a numerical value.So, moving on, the first answer is approximately 498,761 products sold.Now, the second question is about the influencer's expected revenue. He gets a 10% commission on each product sold. So, each product is R200, so his commission per product is 10% of 200, which is R20.Therefore, his total revenue is the number of products sold multiplied by R20.So, 498,761 * 20 = ?Let me compute that.498,761 * 20 = 9,975,220.So, approximately R9,975,220.But let me check if I should use the exact value or if I need to consider more decimal places.Wait, the number of products sold was approximately 498,760.6425, so multiplying by 20 gives 9,975,212.85. So, approximately R9,975,213.But again, since we're dealing with money, we might need to round to the nearest whole number, so R9,975,213.Alternatively, if we use the exact value of C(500,000), which is 1 / (1 + e^{-6}), then the expected number of sales is 500,000 / (1 + e^{-6}).So, the exact revenue would be 500,000 / (1 + e^{-6}) * 20.But I think for the purposes of this problem, the approximate value is sufficient.Wait, but let me make sure I didn't make a mistake in interpreting the function. The function C(x) is the conversion rate for each follower, so the expected number of sales is indeed 500,000 * C(500,000). That seems correct.Alternatively, sometimes in marketing, the conversion rate might be applied to the number of impressions or something else, but in this case, the problem states that C(x) is the probability of conversion for each follower, so it's per person.Therefore, the calculations seem correct.So, summarizing:1. Expected number of products sold: Approximately 498,761.2. Expected revenue: Approximately R9,975,213.But let me write the exact expressions as well, in case the problem expects it.For the first part, the exact expected number is 500,000 / (1 + e^{-6}).For the second part, it's 500,000 / (1 + e^{-6}) * 20.But since e^{-6} is a known value, we can compute it more precisely.Let me compute e^{-6} more accurately.e^{-6} ≈ 0.0024787521766663585.So, 1 + e^{-6} ≈ 1.0024787521766663.Therefore, 500,000 / 1.0024787521766663 ≈ ?Let me compute that.500,000 / 1.0024787521766663 ≈ 500,000 * (1 / 1.0024787521766663).1 / 1.0024787521766663 ≈ 0.997521285.So, 500,000 * 0.997521285 ≈ 498,760.6425.So, approximately 498,760.64, which we can round to 498,761.Similarly, the revenue is 498,760.64 * 20 = 9,975,212.8, which is approximately R9,975,213.Alternatively, if we keep more decimal places, it's R9,975,212.80, but since we're dealing with whole products and whole currency units, rounding to the nearest whole number is appropriate.Therefore, the answers are:1. Approximately 498,761 products sold.2. Approximately R9,975,213 in revenue.Wait, but let me check if I should present the answers in a specific format, like using commas as thousand separators or something. In Brazil, they use commas as decimal separators and periods for thousands, but in this case, since the user wrote R200,000, I think they use commas as thousand separators. So, 498,761 would be 498.761, but in the answer, I think it's better to use the international format with commas as thousand separators.So, 498,761 and R9,975,213.Alternatively, maybe the problem expects the answers in terms of e^{-6}, but I think the numerical approximation is fine.Wait, another thought: Is the conversion rate per follower, or is it the total conversion rate? Because sometimes, in marketing, the conversion rate can be expressed as the total number of conversions divided by the total number of impressions or visitors. But in this case, the problem states that C(x) is the probability of conversion of a follower into a buyer. So, it's per follower.Therefore, the expected number of sales is indeed 500,000 * C(500,000).So, I think my approach is correct.Another way to think about it: If each follower has a probability C(x) of buying, then the expected number of buyers is the sum over all followers of C(x). Since all followers are identical in this model, it's just 500,000 * C(500,000).Yes, that makes sense.Therefore, I'm confident with my calculations.</think>

🧠:<think>Okay, so I have this problem about an elderly resident living next to a nightclub. The resident measures the sound intensity with a decibel meter, and the level varies over time according to the function ( L(t) = 70 + 10 sinleft(frac{pi t}{3}right) ). There are two parts to the problem: first, finding the average sound intensity level over a 9-hour night, and second, determining how long the sound exceeds 75 dB during that time.Starting with part 1: calculating the average sound intensity level. I remember that the average value of a function over an interval [a, b] is given by the integral of the function from a to b divided by the length of the interval. So, in this case, the average ( bar{L} ) would be:[bar{L} = frac{1}{9 - 0} int_{0}^{9} L(t) , dt]Substituting the given function into the integral:[bar{L} = frac{1}{9} int_{0}^{9} left(70 + 10 sinleft(frac{pi t}{3}right)right) dt]I can split this integral into two parts:[bar{L} = frac{1}{9} left( int_{0}^{9} 70 , dt + int_{0}^{9} 10 sinleft(frac{pi t}{3}right) dt right)]Calculating the first integral:[int_{0}^{9} 70 , dt = 70t Big|_{0}^{9} = 70 times 9 - 70 times 0 = 630]Now, the second integral:[int_{0}^{9} 10 sinleft(frac{pi t}{3}right) dt]I need to find the antiderivative of ( sinleft(frac{pi t}{3}right) ). The integral of ( sin(ax) ) is ( -frac{1}{a} cos(ax) ), so applying that here:Let ( a = frac{pi}{3} ), so the integral becomes:[10 times left( -frac{3}{pi} cosleft(frac{pi t}{3}right) right) Big|_{0}^{9}]Simplifying:[- frac{30}{pi} left[ cosleft(frac{pi t}{3}right) Big|_{0}^{9} right]]Calculating the cosine terms at the limits:At ( t = 9 ):[cosleft(frac{pi times 9}{3}right) = cos(3pi) = -1]At ( t = 0 ):[cosleft(frac{pi times 0}{3}right) = cos(0) = 1]So, substituting back:[- frac{30}{pi} [ (-1) - (1) ] = - frac{30}{pi} (-2) = frac{60}{pi}]Therefore, the second integral is ( frac{60}{pi} ).Putting it all together for the average:[bar{L} = frac{1}{9} left( 630 + frac{60}{pi} right ) = frac{630}{9} + frac{60}{9pi} = 70 + frac{20}{3pi}]Calculating ( frac{20}{3pi} ) approximately, since ( pi approx 3.1416 ):[frac{20}{3 times 3.1416} approx frac{20}{9.4248} approx 2.122]So, the average sound intensity level is approximately ( 70 + 2.122 = 72.122 ) dB. But since the question asks for the exact value, I should leave it in terms of pi:[bar{L} = 70 + frac{20}{3pi} text{ dB}]Wait, actually, let me double-check my calculations. The integral of the sine function gave me ( frac{60}{pi} ), and when I divide by 9, it's ( frac{60}{9pi} = frac{20}{3pi} ). Yes, that seems correct. So, the exact average is ( 70 + frac{20}{3pi} ) dB.Moving on to part 2: determining the total duration when ( L(t) > 75 ) dB. So, I need to solve the inequality:[70 + 10 sinleft(frac{pi t}{3}right) > 75]Subtracting 70 from both sides:[10 sinleft(frac{pi t}{3}right) > 5]Divide both sides by 10:[sinleft(frac{pi t}{3}right) > 0.5]So, I need to find all ( t ) in [0, 9] such that ( sinleft(frac{pi t}{3}right) > 0.5 ).I know that ( sin(theta) > 0.5 ) when ( theta ) is in the intervals ( (frac{pi}{6} + 2pi k, frac{5pi}{6} + 2pi k) ) for integer ( k ).So, let me set ( theta = frac{pi t}{3} ), which means:[frac{pi}{6} < frac{pi t}{3} < frac{5pi}{6}]Solving for ( t ):Multiply all parts by ( frac{3}{pi} ):[frac{1}{2} < t < frac{5}{2}]So, in each period of the sine function, the sound exceeds 75 dB for ( frac{5}{2} - frac{1}{2} = 2 ) hours.Now, I need to determine how many such periods occur within the 9-hour interval.The period ( T ) of ( sinleft(frac{pi t}{3}right) ) is ( frac{2pi}{pi/3} = 6 ) hours. So, the function repeats every 6 hours.In 9 hours, there is 1 full period (6 hours) and an additional 3 hours.In each full period, the sound exceeds 75 dB for 2 hours. So, in 6 hours, it's 2 hours.Now, for the remaining 3 hours (from t=6 to t=9), I need to check if the sine function exceeds 0.5.Let me analyze the function from t=6 to t=9.At t=6:[sinleft(frac{pi times 6}{3}right) = sin(2pi) = 0]At t=9:[sinleft(frac{pi times 9}{3}right) = sin(3pi) = 0]So, the sine function goes from 0 at t=6, up to 1 at t=7.5, and back to 0 at t=9.So, in this interval, the sine function will exceed 0.5 twice: once when increasing from 0 to 1, and once when decreasing from 1 to 0.Wait, actually, in the interval t=6 to t=9, the function is symmetric around t=7.5.So, let's find the points where ( sinleft(frac{pi t}{3}right) = 0.5 ) in this interval.Set ( sinleft(frac{pi t}{3}right) = 0.5 ):[frac{pi t}{3} = frac{pi}{6} + 2pi k quad text{or} quad frac{5pi}{6} + 2pi k]Solving for ( t ):First solution:[frac{pi t}{3} = frac{pi}{6} implies t = frac{3}{pi} times frac{pi}{6} = frac{1}{2} = 0.5]But t=0.5 is before t=6, so not in our interval.Second solution:[frac{pi t}{3} = frac{5pi}{6} implies t = frac{3}{pi} times frac{5pi}{6} = frac{15}{6} = 2.5]Again, t=2.5 is before t=6, so not in our interval.Wait, maybe I need to consider the next period. Since the period is 6 hours, the next occurrence after t=6 would be at t=6 + 0.5 = 6.5 and t=6 + 2.5 = 8.5.Wait, let's think differently. The function ( sinleft(frac{pi t}{3}right) ) has a period of 6, so from t=6 to t=9, it's the same as from t=0 to t=3.So, in t=0 to t=3, when does ( sinleft(frac{pi t}{3}right) > 0.5 )?Set ( sinleft(frac{pi t}{3}right) = 0.5 ):[frac{pi t}{3} = frac{pi}{6} implies t = 0.5][frac{pi t}{3} = frac{5pi}{6} implies t = 2.5]So, in t=0 to t=3, the function is above 0.5 between t=0.5 and t=2.5, which is 2 hours.Therefore, in t=6 to t=9, the function is above 0.5 between t=6.5 and t=8.5, which is another 2 hours.Wait, but hold on. From t=6 to t=9, the function goes from 0 up to 1 at t=7.5 and back to 0. So, the function crosses 0.5 at t=6.5 and t=8.5.So, the duration when L(t) > 75 dB in t=6 to t=9 is from 6.5 to 8.5, which is 2 hours.Therefore, in total, over 9 hours, we have:- From t=0 to t=6: 2 hours above 75 dB- From t=6 to t=9: 2 hours above 75 dBWait, but hold on. From t=0 to t=6, the function exceeds 75 dB for 2 hours, and from t=6 to t=9, it also exceeds for 2 hours. So, total duration is 4 hours.But wait, let me verify this because sometimes when functions are periodic, overlapping can occur, but in this case, since the periods are non-overlapping, it should be additive.Wait, actually, no. From t=0 to t=6, the function exceeds 75 dB for 2 hours. Then, from t=6 to t=9, it's another 2 hours. So, total is 4 hours.But let me double-check.In the first period, t=0 to t=6, the function exceeds 75 dB from t=0.5 to t=2.5, which is 2 hours.Then, in the next period, t=6 to t=12, but we only go up to t=9. So, in t=6 to t=9, the function exceeds 75 dB from t=6.5 to t=8.5, which is another 2 hours.So, total duration is 2 + 2 = 4 hours.Wait, but let me think again. The function is periodic with period 6, so in each 6-hour block, it exceeds 75 dB for 2 hours. So, in 9 hours, which is 1 full period plus 3 extra hours, we have 2 hours from the first period and 2 hours from the partial period, totaling 4 hours.But wait, in the partial period from t=6 to t=9, the function exceeds 75 dB for 2 hours, as we saw. So, yes, 4 hours in total.But let me confirm by graphing or testing specific points.At t=0: L(t)=70 + 10 sin(0)=70 dBAt t=0.5: L(t)=70 + 10 sin(π/6)=70 + 5=75 dBAt t=1: L(t)=70 + 10 sin(π/3)=70 + 10*(√3/2)=70 + 8.66≈78.66 dBAt t=2: L(t)=70 + 10 sin(2π/3)=70 + 10*(√3/2)=78.66 dBAt t=2.5: L(t)=70 + 10 sin(5π/6)=70 + 5=75 dBAt t=3: L(t)=70 + 10 sin(π)=70 + 0=70 dBAt t=4: L(t)=70 + 10 sin(4π/3)=70 + 10*(-√3/2)=70 -8.66≈61.34 dBAt t=5: L(t)=70 + 10 sin(5π/3)=70 + 10*(-√3/2)=61.34 dBAt t=6: L(t)=70 + 10 sin(2π)=70 dBSo, from t=0.5 to t=2.5, it's above 75 dB, which is 2 hours.Similarly, from t=6.5 to t=8.5, let's check:At t=6.5: sin(π*6.5/3)=sin(13π/6)=sin(π/6)=0.5, so L(t)=75 dBAt t=7: sin(7π/3)=sin(π/3)=√3/2≈0.866, so L(t)=70 + 8.66≈78.66 dBAt t=8: sin(8π/3)=sin(2π/3)=√3/2≈0.866, so L(t)=78.66 dBAt t=8.5: sin(8.5π/3)=sin(17π/6)=sin(π/6)=0.5, so L(t)=75 dBSo, from t=6.5 to t=8.5, it's above 75 dB, which is another 2 hours.Therefore, total duration is 2 + 2 = 4 hours.Wait, but hold on. The period is 6 hours, so from t=0 to t=6, it's 2 hours above 75 dB. From t=6 to t=9, which is 3 hours, the function goes from 0 up to 1 and back to 0. So, it crosses 0.5 at t=6.5 and t=8.5, so the duration is 2 hours in that interval as well.Therefore, total duration is 4 hours.But wait, let me think again. The function is symmetric, so in each period, it's above 75 dB for 2 hours. So, in 9 hours, which is 1.5 periods, we have 2 hours per period, so 2 * 1.5 = 3 hours? Wait, that contradicts my earlier conclusion.Wait, no. Because in the first period (0-6), it's 2 hours. In the next 3 hours (6-9), it's another 2 hours. So, total is 4 hours. But 9 hours is 1.5 periods, so 1.5 * 2 = 3 hours? Hmm, conflicting conclusions.Wait, perhaps I need to calculate the exact times when L(t) >75 dB in the interval [0,9].Let me solve the inequality ( sinleft(frac{pi t}{3}right) > 0.5 ) for t in [0,9].The general solution for ( sin(theta) > 0.5 ) is ( theta in (frac{pi}{6} + 2pi k, frac{5pi}{6} + 2pi k) ) for integer k.So, ( frac{pi t}{3} in (frac{pi}{6} + 2pi k, frac{5pi}{6} + 2pi k) )Solving for t:Multiply all parts by ( frac{3}{pi} ):( t in left( frac{1}{2} + 6k, frac{5}{2} + 6k right) )Now, find all intervals within [0,9]:For k=0: ( t in (0.5, 2.5) )For k=1: ( t in (6.5, 8.5) )For k=2: ( t in (12.5, 14.5) ) which is outside our interval.So, the intervals where L(t) >75 dB are (0.5, 2.5) and (6.5, 8.5). Each interval is 2 hours long, so total duration is 4 hours.Therefore, the total duration is 4 hours.Wait, but earlier I thought 1.5 periods would give 3 hours, but that's incorrect because the function doesn't exceed 75 dB for 2 hours in each period, but rather in each period, it's 2 hours, regardless of the period length. So, in 9 hours, which includes 1 full period (6 hours) and 3 extra hours, we have 2 hours from the first period and 2 hours from the partial period, totaling 4 hours.Yes, that makes sense. So, the total duration is 4 hours.But let me just confirm by integrating or another method.Alternatively, I can calculate the total time by finding the measure of t where ( sinleft(frac{pi t}{3}right) > 0.5 ) in [0,9].As we found, the solutions are t in (0.5, 2.5) and (6.5,8.5). Each interval is 2 hours, so total is 4 hours.Therefore, the answer is 4 hours.But wait, let me think again. The function is periodic with period 6, so in each 6-hour block, it's above 75 dB for 2 hours. So, in 9 hours, which is 1 full period plus 3 hours, we have 2 hours from the first period and in the next 3 hours, the function is above 75 dB for 2 hours (from 6.5 to 8.5). So, total is 4 hours.Yes, that seems consistent.So, summarizing:1. The average sound intensity level is ( 70 + frac{20}{3pi} ) dB.2. The total duration when the sound exceeds 75 dB is 4 hours.I think that's it.</think>

🧠:<think>Okay, so I have this problem about RSA cryptography, and I need to figure it out step by step. Let me start by recalling what I know about RSA. RSA is a public-key cryptosystem, which means it uses two keys: a public key for encryption and a private key for decryption. The security of RSA relies on the difficulty of factoring large prime numbers.The problem has two parts. The first part is to calculate the public key (n, e) and the private key (d) given two primes p = 11 and q = 13, and an encryption exponent e = 7. The second part is to encrypt a message m = 9 using the public key and then decrypt it back using the private key to verify that it matches the original message.Starting with part 1: Calculating the public and private keys.First, the public key n is the product of the two primes p and q. So, n = p * q. Let me compute that.n = 11 * 13. Hmm, 11 times 13 is 143. So, n is 143. Got that.Next, I need to compute the totient function φ(n). For two distinct primes p and q, φ(n) is equal to (p - 1)(q - 1). So, φ(n) = (11 - 1)(13 - 1) = 10 * 12. Let me calculate that: 10 times 12 is 120. So, φ(n) is 120.Now, the encryption exponent e is given as 7. The private key d must satisfy the congruence e * d ≡ 1 mod φ(n). In other words, d is the multiplicative inverse of e modulo φ(n). So, I need to find a number d such that when I multiply it by 7, the result is congruent to 1 modulo 120.To find d, I can use the Extended Euclidean Algorithm, which finds integers x and y such that ax + by = gcd(a, b). In this case, a is 7 and b is 120. Since 7 and 120 are coprime (their greatest common divisor is 1), there exist integers x and y such that 7x + 120y = 1. The x here will be the multiplicative inverse of 7 modulo 120, which is our d.Let me apply the Extended Euclidean Algorithm step by step.First, divide 120 by 7:120 ÷ 7 = 17 with a remainder of 1, because 7*17 = 119, and 120 - 119 = 1.So, we can write this as:120 = 7*17 + 1Now, rearranging this equation to solve for 1:1 = 120 - 7*17This equation shows that 1 is a linear combination of 120 and 7, which confirms that their gcd is 1. From this, we can see that x is -17 and y is 1, because 7*(-17) + 120*1 = 1.However, we need d to be a positive integer less than φ(n), which is 120. So, we take x = -17 and add 120 to it until we get a positive number.-17 + 120 = 103. So, d is 103.Let me verify that 7 * 103 ≡ 1 mod 120.Calculating 7 * 103: 7*100 = 700, 7*3 = 21, so 700 + 21 = 721.Now, divide 721 by 120 to find the remainder. 120*6 = 720, so 721 - 720 = 1. Therefore, 721 mod 120 is 1. So, yes, 7*103 ≡ 1 mod 120. That checks out.So, the public key is (n, e) = (143, 7), and the private key is d = 103.Moving on to part 2: Encrypting the message m = 9 using the public key and then decrypting it.First, encryption. The encryption formula in RSA is: c = m^e mod n. So, c = 9^7 mod 143.Calculating 9^7 seems a bit tedious, but I can compute it step by step, taking modulus at each step to keep the numbers manageable.Let me compute 9^2 first: 9*9 = 81.Then 9^3 = 9^2 * 9 = 81*9. Let's compute 81*9: 80*9 = 720, 1*9 = 9, so 720 + 9 = 729. Now, 729 mod 143. Let's see how many times 143 goes into 729.143*5 = 715. 729 - 715 = 14. So, 729 mod 143 is 14. So, 9^3 mod 143 is 14.Next, 9^4 = 9^3 * 9 = 14*9 = 126. 126 mod 143 is 126, since 126 < 143.9^5 = 9^4 * 9 = 126*9. Let's compute 126*9: 100*9=900, 20*9=180, 6*9=54. So, 900 + 180 = 1080, 1080 + 54 = 1134. Now, 1134 mod 143.Let's divide 1134 by 143. 143*7 = 1001. 1134 - 1001 = 133. So, 1134 mod 143 is 133.So, 9^5 mod 143 is 133.9^6 = 9^5 * 9 = 133*9. Let's compute 133*9: 100*9=900, 30*9=270, 3*9=27. So, 900 + 270 = 1170, 1170 + 27 = 1197. Now, 1197 mod 143.Divide 1197 by 143: 143*8 = 1144. 1197 - 1144 = 53. So, 1197 mod 143 is 53. Therefore, 9^6 mod 143 is 53.9^7 = 9^6 * 9 = 53*9. 53*9: 50*9=450, 3*9=27, so 450 + 27 = 477. Now, 477 mod 143.143*3 = 429. 477 - 429 = 48. So, 477 mod 143 is 48. Therefore, 9^7 mod 143 is 48.So, the ciphertext c is 48.Now, to decrypt the message, we use the private key d = 103. The decryption formula is: m = c^d mod n. So, m = 48^103 mod 143.Calculating 48^103 mod 143 directly seems impossible because the exponent is so large. Instead, I can use the method of exponentiation by squaring, which allows me to compute large exponents modulo n efficiently.But before that, maybe I can find a pattern or use Euler's theorem. Since φ(143) = 120, and 48 and 143 are coprime (since 48 is less than 143 and factors of 48 are 2, 3, which don't divide 143), Euler's theorem tells us that 48^120 ≡ 1 mod 143.Therefore, 48^103 can be written as 48^(120 - 17) = 48^(-17) mod 143. But that might not be helpful. Alternatively, since 103 = 64 + 32 + 4 + 2 + 1, I can compute 48^1, 48^2, 48^4, 48^8, 48^16, 48^32, 48^64 mod 143, and then multiply the relevant ones together.Let me proceed step by step.Compute 48^1 mod 143: that's just 48.Compute 48^2 mod 143: 48*48 = 2304. Now, 2304 ÷ 143: 143*16 = 2288. 2304 - 2288 = 16. So, 48^2 mod 143 is 16.Compute 48^4 mod 143: (48^2)^2 mod 143 = 16^2 mod 143. 16*16 = 256. 256 mod 143: 143*1 = 143, 256 - 143 = 113. So, 48^4 mod 143 is 113.Compute 48^8 mod 143: (48^4)^2 mod 143 = 113^2 mod 143. 113*113: Let's compute 100*100=10000, 100*13=1300, 13*100=1300, 13*13=169. Wait, that's not the right way. Alternatively, 113*113: 100*100=10000, 100*13=1300, 13*100=1300, 13*13=169. So, adding up: 10000 + 1300 + 1300 + 169 = 12769. Now, 12769 mod 143.Let me compute how many times 143 goes into 12769. 143*80 = 11440. 12769 - 11440 = 1329. 143*9 = 1287. 1329 - 1287 = 42. So, 12769 mod 143 is 42. Therefore, 48^8 mod 143 is 42.Compute 48^16 mod 143: (48^8)^2 mod 143 = 42^2 mod 143. 42*42 = 1764. 1764 ÷ 143: 143*12 = 1716. 1764 - 1716 = 48. So, 48^16 mod 143 is 48.Compute 48^32 mod 143: (48^16)^2 mod 143 = 48^2 mod 143, which we already computed as 16.Compute 48^64 mod 143: (48^32)^2 mod 143 = 16^2 mod 143, which is 256 mod 143. As before, 256 - 143 = 113. So, 48^64 mod 143 is 113.Now, we have all the necessary powers:48^1 mod 143 = 4848^2 mod 143 = 1648^4 mod 143 = 11348^8 mod 143 = 4248^16 mod 143 = 4848^32 mod 143 = 1648^64 mod 143 = 113Now, 103 in binary is 64 + 32 + 4 + 2 + 1. So, 103 = 64 + 32 + 4 + 2 + 1. Therefore, 48^103 mod 143 = (48^64 * 48^32 * 48^4 * 48^2 * 48^1) mod 143.Let me compute each multiplication step by step, taking modulus 143 at each step.First, multiply 48^64 and 48^32: 113 * 16 mod 143.113 * 16: 100*16=1600, 13*16=208. So, 1600 + 208 = 1808. 1808 mod 143.143*12 = 1716. 1808 - 1716 = 92. So, 113*16 mod 143 is 92.Next, multiply this result by 48^4: 92 * 113 mod 143.92 * 113: Let's compute 90*113=10170, 2*113=226. So, 10170 + 226 = 10396. Now, 10396 mod 143.Let me compute 143*72 = 10296. 10396 - 10296 = 100. So, 92*113 mod 143 is 100.Next, multiply this result by 48^2: 100 * 16 mod 143.100 * 16 = 1600. 1600 mod 143: 143*11 = 1573. 1600 - 1573 = 27. So, 100*16 mod 143 is 27.Next, multiply this result by 48^1: 27 * 48 mod 143.27 * 48: 20*48=960, 7*48=336. 960 + 336 = 1296. 1296 mod 143.143*9 = 1287. 1296 - 1287 = 9. So, 27*48 mod 143 is 9.Therefore, after all these multiplications, we end up with 9. So, 48^103 mod 143 is 9. Which is our original message m. So, the decryption works correctly.Wait, let me recap the steps to ensure I didn't make a mistake:1. 48^64 * 48^32 = 113 * 16 = 1808 mod 143 = 922. 92 * 48^4 = 92 * 113 = 10396 mod 143 = 1003. 100 * 48^2 = 100 * 16 = 1600 mod 143 = 274. 27 * 48^1 = 27 * 48 = 1296 mod 143 = 9Yes, that seems correct. So, the decrypted message is indeed 9, which matches the original message. So, the encryption and decryption processes worked as expected.Let me just double-check the encryption step as well, in case I made an error there.Encryption: m = 9, e = 7, n = 143.Compute 9^7 mod 143.Earlier, I computed it step by step:9^2 = 819^3 = 9^2 * 9 = 81*9 = 729 mod 143 = 149^4 = 14*9 = 1269^5 = 126*9 = 1134 mod 143 = 1339^6 = 133*9 = 1197 mod 143 = 539^7 = 53*9 = 477 mod 143 = 48So, c = 48.Then, decrypting c = 48 with d = 103 gives back m = 9. So, everything checks out.I think I've covered all the steps thoroughly. I used the Extended Euclidean Algorithm to find the private key, performed modular exponentiation for both encryption and decryption, and verified that the decrypted message is the same as the original. It seems solid.One thing I might have overlooked is ensuring that all the intermediate steps in the decryption process were correct. Let me quickly recheck the multiplications:- 113 * 16 = 1808; 1808 - 143*12 = 1808 - 1716 = 92. Correct.- 92 * 113: 92*100=9200, 92*13=1196; 9200 + 1196 = 10396. 10396 - 143*72 = 10396 - 10296 = 100. Correct.- 100 * 16 = 1600; 1600 - 143*11 = 1600 - 1573 = 27. Correct.- 27 * 48 = 1296; 1296 - 143*9 = 1296 - 1287 = 9. Correct.Yes, all steps are accurate. So, I can confidently say that the public key is (143, 7), the private key is 103, the ciphertext is 48, and decrypting 48 gives back 9.Final Answer1. The public key is ( boxed{(143, 7)} ) and the private key is ( boxed{103} ).2. The encrypted message is ( boxed{48} ) and the decrypted message is ( boxed{9} ).</think>