🧠:<think>Okay, so I have this problem about a broadcasting network and their compliance with advertising regulations. Let me try to figure this out step by step.First, the problem says that the average number of advertisements per hour over any 24-hour period shouldn't exceed 12. The lawyer has a model for the number of ads per hour, which is given by the function ( f(t) = 3sin(pi t/12) + 9 ) where ( t ) is in hours from 0 to 24.I need to find the average number of ads per hour over 24 hours. Hmm, I remember that the average value of a function over an interval [a, b] is given by the integral of the function over that interval divided by the length of the interval. So, the formula should be:[text{Average} = frac{1}{b - a} int_{a}^{b} f(t) , dt]In this case, ( a = 0 ) and ( b = 24 ), so the average number of ads per hour is:[text{Average} = frac{1}{24 - 0} int_{0}^{24} (3sin(pi t/12) + 9) , dt]Alright, let me compute this integral. I can split the integral into two parts:[int_{0}^{24} 3sinleft(frac{pi t}{12}right) , dt + int_{0}^{24} 9 , dt]Let me compute the first integral:[int 3sinleft(frac{pi t}{12}right) , dt]I need to find the antiderivative of ( 3sinleft(frac{pi t}{12}right) ). The integral of ( sin(ax) ) is ( -frac{1}{a}cos(ax) ), so applying that here:Let ( a = frac{pi}{12} ), so the integral becomes:[3 times left( -frac{12}{pi} cosleft(frac{pi t}{12}right) right) = -frac{36}{pi} cosleft(frac{pi t}{12}right) + C]Now, evaluating this from 0 to 24:At ( t = 24 ):[-frac{36}{pi} cosleft(frac{pi times 24}{12}right) = -frac{36}{pi} cos(2pi) = -frac{36}{pi} times 1 = -frac{36}{pi}]At ( t = 0 ):[-frac{36}{pi} cosleft(0right) = -frac{36}{pi} times 1 = -frac{36}{pi}]So, subtracting the lower limit from the upper limit:[-frac{36}{pi} - left(-frac{36}{pi}right) = 0]Wait, that's interesting. The integral of the sine function over a full period is zero. That makes sense because the sine function is symmetric and positive and negative areas cancel out over a full period.So, the first integral is zero.Now, the second integral:[int_{0}^{24} 9 , dt = 9t bigg|_{0}^{24} = 9 times 24 - 9 times 0 = 216]So, putting it all together, the total integral is 0 + 216 = 216.Therefore, the average number of ads per hour is:[frac{216}{24} = 9]Wait, 9? That's below the 12 limit. So, the network is in compliance because the average is 9, which is less than 12.But let me double-check my calculations because sometimes I might make a mistake.First, the integral of the sine function:Yes, over 0 to 24, since the period of ( sin(pi t / 12) ) is ( 24 ) hours, so over one full period, the integral is zero. So, that part is correct.The integral of 9 from 0 to 24 is indeed 9*24=216. Divided by 24 gives 9. So, that's correct.So, the average is 9, which is within the 12 limit. So, they are compliant.Now, moving on to the second part. Suppose the network can adjust the number of ads each hour by a constant amount ( k ). So, the new function would be ( f(t) + k = 3sin(pi t /12) + 9 + k ).We need to find the maximum value of ( k ) such that the average over any 24-hour period doesn't exceed 12.Wait, but the average is calculated over any 24-hour period. So, does that mean we have to ensure that for any interval of 24 hours, the average doesn't exceed 12? Or is it just over the 24-hour period as a whole?Wait, the regulation says "over any 24-hour period," so it's not just the average over the specific 24-hour period we considered, but any consecutive 24-hour period.Hmm, that complicates things a bit. Because if the function is periodic, then shifting the interval might not change the average, but if it's not, it could.Wait, in our case, the function ( f(t) = 3sin(pi t /12) + 9 ) is periodic with period 24 hours, right? Because the sine function has a period of ( 2pi / (pi /12) ) = 24 ). So, the function repeats every 24 hours.Therefore, the average over any 24-hour period would be the same as the average over any other 24-hour period. So, the average is always 9, regardless of where you start the 24-hour period.Therefore, if we add a constant ( k ), the average would just increase by ( k ). So, the new average would be ( 9 + k ).To comply with the regulation, we need ( 9 + k leq 12 ). Therefore, ( k leq 3 ).So, the maximum value of ( k ) is 3.Wait, but let me think again. Is the function really periodic with period 24? Let's check.The function is ( 3sin(pi t /12) + 9 ). The sine function has a period of ( 2pi / (pi /12) ) = 24 ). So, yes, it's periodic with period 24. Therefore, the average over any 24-hour interval is the same.Therefore, adding a constant ( k ) would just shift the average by ( k ). So, the maximum ( k ) is 3.But wait, let me make sure. Suppose ( k = 3 ), then the function becomes ( 3sin(pi t /12) + 12 ). The average would be 12, which is exactly the limit. So, that's acceptable.If ( k ) were more than 3, say 4, then the average would be 13, which exceeds the limit.Therefore, the maximum ( k ) is 3.But hold on, is there any other consideration? For example, does the instantaneous number of ads ever exceed some limit? Or is it only the average?The problem statement only mentions the average number of advertisements per hour over any 24-hour period. So, as long as the average doesn't exceed 12, they are compliant. So, even if at some points the number of ads is higher, as long as the average is okay, it's fine.Therefore, the maximum ( k ) is 3.But let me think again about the first part. The average is 9, so they can add up to 3 more on average without exceeding the limit.Yes, that seems correct.So, to recap:1. The average number of ads per hour is 9, so they are compliant.2. The maximum ( k ) they can add is 3, making the average 12, which is the limit.I think that's solid.Final Answer1. The average number of advertisements per hour is boxed{9}, so the network is in compliance.2. The maximum value of ( k ) is boxed{3}.</think>

🧠:<think>First, I'll tackle the first problem where the elderly woman and her 9 friends form a circle when she was 10 years old. There are a total of 10 children, and each occupies 0.5 meters along the circumference. To find the total circumference, I'll multiply the number of children by the space each occupies: 10 * 0.5 = 5 meters. Using the circumference formula ( C = 2pi r ), I can solve for the radius ( r ) by rearranging the formula: ( r = frac{C}{2pi} ). Plugging in the numbers, ( r = frac{5}{2pi} ), which gives approximately 0.796 meters.Next, for the second problem, the woman wants to include herself in the circle with her 20 grandchildren, making a total of 21 people. Each person now occupies 0.4 meters. The total circumference will be 21 * 0.4 = 8.4 meters. Using the same circumference formula, I'll solve for the new radius: ( r = frac{8.4}{2pi} ), which calculates to approximately 1.336 meters.</think>

🧠:<think>Okay, so I have two problems here about archery and probability. Let me tackle them one by one.Problem 1: Probability of landing in the 6th scoring zoneAlright, the target has 10 concentric scoring zones, each with radius ( r_n = frac{n}{10} ) meters. So, the 6th zone is between ( r = frac{5}{10} = 0.5 ) meters and ( r = frac{6}{10} = 0.6 ) meters. The probability density function (pdf) given is ( f(r) = 3r^2 ) for ( 0 leq r leq 1 ).I remember that the probability of an event in a continuous distribution is the integral of the pdf over the interval of interest. So, the probability that the arrow lands in the 6th zone is the integral of ( f(r) ) from 0.5 to 0.6.Let me write that down:( P(0.5 leq r leq 0.6) = int_{0.5}^{0.6} 3r^2 , dr )To compute this integral, I can find the antiderivative of ( 3r^2 ). The antiderivative of ( r^2 ) is ( frac{r^3}{3} ), so multiplying by 3 gives ( r^3 ).So, evaluating from 0.5 to 0.6:( P = [r^3]_{0.5}^{0.6} = (0.6)^3 - (0.5)^3 )Calculating each term:( (0.6)^3 = 0.216 )( (0.5)^3 = 0.125 )Subtracting:( 0.216 - 0.125 = 0.091 )So, the probability is 0.091, which is 9.1%.Wait, let me double-check the calculations:( 0.6^3 = 0.6 * 0.6 * 0.6 = 0.36 * 0.6 = 0.216 )( 0.5^3 = 0.125 )Yes, 0.216 - 0.125 is indeed 0.091. So, that seems correct.Problem 2: Probability of hitting the bullseye at least once in 5 attemptsChristopher Davis hits the bullseye 70% of the time, but the aspiring archer has a probability ( p ) of hitting the bullseye each time. The archer wants a 90% chance of hitting the bullseye at least once in 5 attempts. We need to find ( p ).Hmm, okay. So, this is a binomial probability problem. The probability of at least one success in 5 trials is 1 minus the probability of zero successes.So, the probability of not hitting the bullseye in a single attempt is ( 1 - p ). Therefore, the probability of not hitting it in all 5 attempts is ( (1 - p)^5 ).We want the probability of hitting at least once to be 90%, so:( 1 - (1 - p)^5 = 0.90 )Solving for ( p ):( (1 - p)^5 = 1 - 0.90 = 0.10 )Take the fifth root of both sides:( 1 - p = (0.10)^{1/5} )Calculating ( (0.10)^{1/5} ). Let me think about this. The fifth root of 0.1.I know that ( 0.1 = 10^{-1} ), so ( (10^{-1})^{1/5} = 10^{-1/5} ).Calculating ( 10^{-1/5} ). Since ( 10^{1/5} ) is approximately the fifth root of 10, which is roughly 1.5849 (since 1.5849^5 ≈ 10). Therefore, ( 10^{-1/5} ≈ 1 / 1.5849 ≈ 0.6309 ).So, ( 1 - p ≈ 0.6309 ), which means ( p ≈ 1 - 0.6309 = 0.3691 ).So, approximately 36.91%.Wait, let me verify that calculation because 0.6309^5 should be approximately 0.10.Calculating ( 0.6309^5 ):First, ( 0.6309^2 ≈ 0.6309 * 0.6309 ≈ 0.398 )Then, ( 0.398 * 0.6309 ≈ 0.251 ) (third power)Fourth power: ( 0.251 * 0.6309 ≈ 0.158 )Fifth power: ( 0.158 * 0.6309 ≈ 0.10 )Yes, that checks out. So, ( p ≈ 0.3691 ), which is approximately 36.91%.But let me see if I can compute it more precisely.Alternatively, using logarithms:( (1 - p)^5 = 0.10 )Take natural logarithm on both sides:( 5 ln(1 - p) = ln(0.10) )( ln(1 - p) = frac{ln(0.10)}{5} )Calculate ( ln(0.10) ≈ -2.302585 )So, ( ln(1 - p) ≈ -2.302585 / 5 ≈ -0.460517 )Exponentiate both sides:( 1 - p ≈ e^{-0.460517} )Calculating ( e^{-0.460517} ). Since ( e^{-0.4605} ≈ 1 / e^{0.4605} ).( e^{0.4605} ≈ 1.5849 ) (since ( ln(1.5849) ≈ 0.4605 ))Therefore, ( e^{-0.4605} ≈ 1 / 1.5849 ≈ 0.6309 )So, same result as before. Therefore, ( p ≈ 1 - 0.6309 = 0.3691 ), which is approximately 36.91%.So, the archer needs a probability ( p ) of about 0.3691 or 36.91% per attempt to have a 90% chance of hitting the bullseye at least once in 5 attempts.Wait, but the problem says that Christopher Davis hits the bullseye 70% of the time, but the archer's probability is ( p ). Is there any relation between ( p ) and 70%? Or is 70% just given as context?Looking back: "Christopher Davis, the young archer's idol, has set a record of hitting bullseye 70% of the time. Assume that the aspiring archer hits the bullseye independently with a probability ( p ). The archer wants to have a 90% chance of hitting the bullseye at least once in 5 attempts. Determine the value of ( p )."So, I think the 70% is just context, and the archer's probability is ( p ), which we need to find such that the probability of at least one bullseye in 5 attempts is 90%.So, my previous calculation is correct, ( p ≈ 0.3691 ).But just to be thorough, let's compute ( (1 - p)^5 = 0.10 ), so ( p = 1 - (0.10)^{1/5} ).Calculating ( (0.10)^{1/5} ) more accurately.We can use logarithms or iterative methods.Alternatively, using a calculator approach:We know that ( 0.6^5 = 0.07776 )( 0.63^5 ): Let's compute 0.63^2 = 0.39690.63^3 = 0.3969 * 0.63 ≈ 0.25000.63^4 ≈ 0.2500 * 0.63 ≈ 0.15750.63^5 ≈ 0.1575 * 0.63 ≈ 0.099225That's very close to 0.10. So, ( 0.63^5 ≈ 0.099225 ), which is approximately 0.10.Therefore, ( (1 - p) ≈ 0.63 ), so ( p ≈ 0.37 ).So, 0.37 is a good approximation. Therefore, ( p ≈ 0.37 ).But since 0.63^5 is approximately 0.099225, which is slightly less than 0.10. So, to get exactly 0.10, we need a slightly higher value than 0.63.Let me compute 0.631^5.0.631^2 = 0.631 * 0.631 ≈ 0.3981610.631^3 ≈ 0.398161 * 0.631 ≈ 0.398161 * 0.6 = 0.2388966, plus 0.398161 * 0.031 ≈ 0.012343, total ≈ 0.251240.631^4 ≈ 0.25124 * 0.631 ≈ 0.25124 * 0.6 = 0.150744, plus 0.25124 * 0.031 ≈ 0.007788, total ≈ 0.1585320.631^5 ≈ 0.158532 * 0.631 ≈ 0.158532 * 0.6 = 0.095119, plus 0.158532 * 0.031 ≈ 0.004914, total ≈ 0.100033Wow, that's very close to 0.10. So, ( (0.631)^5 ≈ 0.100033 ), which is just over 0.10. Therefore, ( (1 - p) ≈ 0.631 ), so ( p ≈ 1 - 0.631 = 0.369 ).So, p is approximately 0.369, which is 36.9%.So, rounding to three decimal places, 0.369.But in the problem, do they specify the form of the answer? It just says "determine the value of p". So, probably, we can write it as approximately 0.369 or 36.9%.Alternatively, if we need more precision, but 0.369 is sufficient.Alternatively, if we express it as a fraction, but 0.369 is approximately 369/1000, which is 36.9%.So, I think 0.369 is a good answer.Summary of Thoughts:1. For the first problem, I correctly identified that the probability is the integral of the pdf over the interval [0.5, 0.6]. Calculated the integral, got 0.091, which is 9.1%.2. For the second problem, recognized it as a binomial probability problem. Set up the equation ( 1 - (1 - p)^5 = 0.90 ), solved for ( p ) by taking the fifth root, approximated it numerically, and found ( p ≈ 0.369 ).I think both solutions are solid. I double-checked the calculations, especially for the second problem, by computing the fifth power of 0.631 and saw that it's very close to 0.10, so that confirms the value of ( p ≈ 0.369 ).Final Answer1. The probability is boxed{0.091}.2. The required probability ( p ) is boxed{0.369}.</think>

🧠:<think>Okay, so I need to figure out the total number of feathers in a Peacock Mandala after n layers. Let me start by understanding the problem step by step.First, the Peacock Mandala is constructed in layers. The first layer has just 1 feather. Then, each subsequent layer adds more feathers around the previous ones. Specifically, each layer n (where n is 2 or more) adds 2^(n-1) feathers. Let me write down the number of feathers added at each layer to see the pattern:- Layer 1: 1 feather- Layer 2: 2^(2-1) = 2^1 = 2 feathers- Layer 3: 2^(3-1) = 2^2 = 4 feathers- Layer 4: 2^(4-1) = 2^3 = 8 feathers- And so on...So, each layer n adds 2^(n-1) feathers. That means the number of feathers added at each layer is doubling each time. Now, to find the total number of feathers after n layers, I need to sum up the feathers from each layer. So, the total number of feathers F(n) would be the sum of 1 + 2 + 4 + 8 + ... + 2^(n-1). Wait, that looks like a geometric series. In a geometric series, each term is a constant multiple of the previous term. In this case, each term is multiplied by 2. The general form of a geometric series is a + ar + ar^2 + ... + ar^(n-1), where a is the first term and r is the common ratio.In our case, the first term a is 1, and the common ratio r is 2. So, the sum S of the first n terms of a geometric series is given by S = a*(r^n - 1)/(r - 1). Plugging in our values, a = 1, r = 2, so S = (2^n - 1)/(2 - 1) = 2^n - 1. Therefore, the total number of feathers F(n) after n layers should be 2^n - 1. Let me test this with the first few layers to make sure.For n = 1: F(1) = 2^1 - 1 = 2 - 1 = 1. That's correct, as the first layer has 1 feather.For n = 2: F(2) = 2^2 - 1 = 4 - 1 = 3. Let's check: Layer 1 has 1, Layer 2 adds 2, so total is 1 + 2 = 3. Correct.For n = 3: F(3) = 2^3 - 1 = 8 - 1 = 7. Checking: 1 + 2 + 4 = 7. Correct.For n = 4: F(4) = 2^4 - 1 = 16 - 1 = 15. Checking: 1 + 2 + 4 + 8 = 15. Correct.Okay, so the formula seems to hold for the first few layers. Therefore, I can be confident that the general formula is F(n) = 2^n - 1.Now, moving on to the second sub-problem: calculating F(10). Using the formula I just derived, F(10) = 2^10 - 1.Calculating 2^10: 2^10 is 1024. So, 1024 - 1 = 1023.Therefore, the total number of feathers in a Peacock Mandala with 10 layers is 1023.Let me just verify that with another approach. The number of feathers added each layer is 1, 2, 4, 8, ..., up to 2^9 for the 10th layer. So, the sum is 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512.Adding these up:1 + 2 = 33 + 4 = 77 + 8 = 1515 + 16 = 3131 + 32 = 6363 + 64 = 127127 + 128 = 255255 + 256 = 511511 + 512 = 1023Yes, that's correct. So, both methods give me 1023 feathers for 10 layers.Therefore, I'm confident that the formula is correct and the calculation for F(10) is accurate.Final Answer1. The general formula for the total number of feathers is boxed{2^n - 1}.2. The total number of feathers in a Peacock Mandala with 10 layers is boxed{1023}.</think>