🧠:<think>Alright, so I have this problem about a fashion event planner organizing a show in an elliptical venue. The first part is to calculate the perimeter of the ellipse using the Ramanujan approximation. The given semi-major axis is 50 meters and the semi-minor axis is 30 meters. The second part is about figuring out how many advertisements can be placed around the perimeter if each is spaced 5 meters apart. Let me tackle this step by step.Starting with the first part: calculating the circumference of the ellipse. I remember that unlike a circle, an ellipse doesn't have a straightforward formula for its perimeter, so approximations are often used. The problem specifically mentions using the Ramanujan approximation, which is given by:[ C approx pi left[ 3(a + b) - sqrt{(3a + b)(a + 3b)} right] ]Okay, so I need to plug in the values of a and b into this formula. Given that a = 50 and b = 30, let me write that down:First, compute ( a + b ). That's 50 + 30, which is 80.Next, compute ( 3(a + b) ). So that's 3 * 80, which equals 240.Now, I need to compute the square root part: ( sqrt{(3a + b)(a + 3b)} ). Let's break that down.Calculate ( 3a + b ): 3*50 + 30 = 150 + 30 = 180.Calculate ( a + 3b ): 50 + 3*30 = 50 + 90 = 140.So, now multiply those two results: 180 * 140. Let me compute that. 180 * 140 is... 180*100=18,000 and 180*40=7,200, so total is 25,200.So, the square root of 25,200. Hmm, let me see. What is sqrt(25,200)? Let's factor that.25,200 can be written as 25,200 = 100 * 252. So sqrt(100*252) = 10*sqrt(252). Now, sqrt(252) can be simplified.252 factors into 4*63, so sqrt(4*63) = 2*sqrt(63). Therefore, sqrt(252) = 2*sqrt(63). So, putting it all together, sqrt(25,200) = 10*2*sqrt(63) = 20*sqrt(63).But wait, sqrt(63) can be simplified further because 63 = 9*7, so sqrt(63) = 3*sqrt(7). Therefore, sqrt(25,200) = 20*3*sqrt(7) = 60*sqrt(7).Hmm, that seems a bit complicated, but maybe I can compute a numerical value for sqrt(7) to approximate this. I know that sqrt(7) is approximately 2.6458.So, sqrt(25,200) ≈ 60 * 2.6458 ≈ 60 * 2.6458. Let me compute that:60 * 2 = 12060 * 0.6458 ≈ 60 * 0.6 = 36 and 60 * 0.0458 ≈ 2.748, so total ≈ 36 + 2.748 = 38.748So, total sqrt(25,200) ≈ 120 + 38.748 ≈ 158.748 meters.Wait, hold on, that can't be right because 158.748 squared is way more than 25,200. Wait, no, actually, 158.748 squared is approximately (160)^2 = 25,600, which is close to 25,200. So, yeah, that seems correct.Wait, but actually, 158.748 squared is approximately 25,200? Let me check:158.748 * 158.748. Let me compute 158 * 158 first. 158 squared is 24,964. Then, 0.748 squared is about 0.559. Then, cross terms: 2*158*0.748 ≈ 2*158*0.75 ≈ 2*118.5 ≈ 237. So, total is approximately 24,964 + 237 + 0.559 ≈ 25,201.559. Wow, that's really close to 25,200. So, sqrt(25,200) ≈ 158.748 meters.So, going back, the square root term is approximately 158.748.Now, plug that back into the Ramanujan formula:C ≈ π [3(a + b) - sqrt((3a + b)(a + 3b))]We have 3(a + b) = 240, and sqrt term ≈ 158.748.So, 240 - 158.748 = 81.252.Therefore, C ≈ π * 81.252.Compute π * 81.252. Since π is approximately 3.1416.So, 81.252 * 3.1416.Let me compute that:First, 80 * 3.1416 = 251.328Then, 1.252 * 3.1416 ≈ Let's compute 1 * 3.1416 = 3.1416, and 0.252 * 3.1416 ≈ 0.252*3 = 0.756, 0.252*0.1416 ≈ 0.0357. So total ≈ 0.756 + 0.0357 ≈ 0.7917.So, 3.1416 + 0.7917 ≈ 3.9333.Therefore, total circumference ≈ 251.328 + 3.9333 ≈ 255.2613 meters.So, approximately 255.26 meters.Wait, let me double-check my calculations because sometimes when I break it down, I might make a mistake.Alternatively, I can compute 81.252 * π directly.81.252 * 3.1416.Compute 80 * 3.1416 = 251.3281.252 * 3.1416:Compute 1 * 3.1416 = 3.14160.252 * 3.1416: Let's compute 0.2 * 3.1416 = 0.628320.05 * 3.1416 = 0.157080.002 * 3.1416 = 0.0062832So, adding those: 0.62832 + 0.15708 = 0.7854, plus 0.0062832 ≈ 0.7916832So, total 0.252 * 3.1416 ≈ 0.7916832Therefore, 1.252 * 3.1416 ≈ 3.1416 + 0.7916832 ≈ 3.9332832Thus, total circumference ≈ 251.328 + 3.9332832 ≈ 255.2612832 meters.So, approximately 255.26 meters.Therefore, the perimeter is approximately 255.26 meters.Wait, but just to make sure, maybe I should check if I computed the square root correctly.Earlier, I had sqrt(25,200) ≈ 158.748.But let me compute 158.748 squared:158.748 * 158.748.Let me compute 158 * 158 = 24,964158 * 0.748 = approx 158 * 0.7 = 110.6, 158 * 0.048 = approx 7.584, so total ≈ 110.6 + 7.584 ≈ 118.184Similarly, 0.748 * 158 = same as above, 118.184And 0.748 * 0.748 ≈ 0.559So, total is 24,964 + 118.184 + 118.184 + 0.559 ≈ 24,964 + 236.368 + 0.559 ≈ 25,200.927, which is very close to 25,200. So, yes, sqrt(25,200) ≈ 158.748 is correct.So, the calculation seems accurate.Therefore, the circumference is approximately 255.26 meters.Now, moving on to the second part: placing advertisements every 5 meters around the perimeter. So, the number of advertisements would be the total perimeter divided by the spacing between each advertisement.So, number of ads ≈ C / 5.Given that C ≈ 255.26 meters, then:Number of ads ≈ 255.26 / 5 ≈ 51.052.Since you can't have a fraction of an advertisement, we need to round this to the nearest whole number. However, in such cases, usually, you can only place whole advertisements, so you might have to see if 51 ads would fit or if you need 52.But let's compute it more accurately.255.26 / 5 = 51.052.So, 51.052 is approximately 51.05, which is just a bit over 51. So, 51 advertisements would cover 51*5=255 meters, leaving a small gap of 0.26 meters. Alternatively, 52 advertisements would require 52*5=260 meters, which is more than the circumference, so that's not possible.Therefore, the maximum number of advertisements that can be placed without exceeding the perimeter is 51.But wait, sometimes, when placing objects around a loop, you can have the last one connect back to the first, so the total length would be exactly the perimeter. So, in that case, if you have n advertisements, each spaced 5 meters apart, the total length would be n*5 meters, which should equal the circumference.But since the circumference is approximately 255.26 meters, which isn't a multiple of 5, you can't have an exact number. So, the number of advertisements would be the integer part of 255.26 / 5, which is 51, with a small leftover space.Alternatively, if the planner wants to space them as evenly as possible, maybe they can adjust the spacing slightly, but the problem says each advertisement must be placed 5 meters apart. So, it's likely that they can only fit 51 advertisements, each 5 meters apart, covering 255 meters, leaving a small gap of about 0.26 meters. Alternatively, if they want to have the last advertisement connect back to the first without overlapping, they might have to reduce the spacing slightly, but the problem specifies 5 meters apart, so I think 51 is the answer.But let me think again. If you have n advertisements, each 5 meters apart, then the total perimeter would be n*5 meters. Since the perimeter is approximately 255.26, n must satisfy n*5 ≈ 255.26, so n ≈ 51.052. Since n must be an integer, the maximum n such that n*5 ≤ 255.26 is 51, because 51*5=255 ≤ 255.26, and 52*5=260 >255.26.Therefore, the number of advertisements is 51.Wait, but sometimes in circular arrangements, the number of intervals is equal to the number of objects, so n advertisements would create n intervals, each of length 5 meters. So, the total circumference would be n*5. Therefore, n = C / 5. So, n ≈ 255.26 /5 ≈51.052. So, n must be 51, as you can't have a fraction.Therefore, the number of advertisements is 51.Alternatively, if the planner wants to have the spacing as close to 5 meters as possible without exceeding, then 51 advertisements would be placed, each 5 meters apart, covering 255 meters, leaving a small gap. Alternatively, if they want to have the spacing exactly 5 meters, they might have to adjust the number to 51, as 52 would require more than the circumference.So, I think the answer is 51 advertisements.But just to make sure, let me re-express the problem:Total perimeter ≈255.26 meters.Each advertisement is spaced 5 meters apart. So, the number of intervals between advertisements is equal to the number of advertisements, because it's a closed loop. So, n advertisements create n intervals. So, n*5 = C. Therefore, n = C /5 ≈255.26 /5≈51.052. Since n must be integer, n=51.Therefore, 51 advertisements can be placed around the perimeter, each 5 meters apart, with a small leftover space of approximately 0.26 meters. Alternatively, if the planner wants to have the spacing exactly 5 meters without any leftover space, they might have to adjust the number, but since 51*5=255 is less than 255.26, it's acceptable.Therefore, the answers are:1. Perimeter ≈255.26 meters.2. Number of advertisements ≈51.But let me check if I made any calculation errors.First, computing 3(a + b): 3*(50+30)=3*80=240. Correct.Then, (3a + b)=150+30=180, (a + 3b)=50+90=140. Correct.Multiplying 180*140=25,200. Correct.sqrt(25,200)=158.748. Correct.Then, 240 -158.748=81.252. Correct.Then, 81.252*π≈81.252*3.1416≈255.26. Correct.Then, 255.26 /5≈51.052≈51. Correct.Yes, all steps seem correct.Therefore, the final answers are:1. The perimeter is approximately 255.26 meters.2. The number of advertisements is 51.Final Answer1. The perimeter of the ellipse is approximately boxed{255.26} meters.2. The number of advertisements that can be placed is boxed{51}.</think>

🧠:<think>Alright, so I have this problem where a retired business executive from South Korea is looking to invest in a local cooperative. There are two projects: a community market expansion and a renewable energy project. I need to figure out the best time to invest in each based on their return models.Starting with the first part, the community market expansion has a rate of return modeled by a quadratic function: ( R(t) = -0.5t^2 + 6t + 12 ). I need to find the time ( t ) at which this rate of return is maximized and also determine what that maximum rate is.Hmm, okay, quadratic functions. I remember that a quadratic function in the form ( at^2 + bt + c ) has its vertex at ( t = -frac{b}{2a} ). Since the coefficient of ( t^2 ) here is negative (-0.5), the parabola opens downward, which means the vertex is the maximum point. So, that should give me the time when the rate of return is the highest.Let me write that down. The formula for the vertex is ( t = -frac{b}{2a} ). In this case, ( a = -0.5 ) and ( b = 6 ). Plugging those into the formula:( t = -frac{6}{2 times -0.5} )Calculating the denominator first: ( 2 times -0.5 = -1 ). So now it's ( t = -frac{6}{-1} ). Dividing two negatives gives a positive, so ( t = 6 ) years.Okay, so the maximum rate of return occurs at 6 years. Now, to find the maximum rate itself, I need to plug ( t = 6 ) back into the original equation ( R(t) ).So, ( R(6) = -0.5(6)^2 + 6(6) + 12 ).Calculating each term step by step:First, ( (6)^2 = 36 ). Then, ( -0.5 times 36 = -18 ).Next, ( 6 times 6 = 36 ).So, putting it all together: ( -18 + 36 + 12 ).Adding those up: ( (-18 + 36) = 18 ), then ( 18 + 12 = 30 ).So, the maximum rate of return is 30%.Wait, that seems pretty high. Let me double-check my calculations.Starting with ( R(6) ):( -0.5 times 36 = -18 ), correct.( 6 times 6 = 36 ), correct.Adding them: ( -18 + 36 = 18 ), then ( 18 + 12 = 30 ). Yeah, that's right. So, 30% is the maximum rate of return at 6 years.Alright, moving on to the second part. The renewable energy project has a return modeled by an exponential function: ( E(t) = 100e^{0.03t} ). I need to find the time ( t ) when the return from this project equals the maximum rate of return from the community market expansion, which we found to be 30%.Wait, hold on. The units here might be different. The community market expansion's rate of return is in percentage, so 30% is 0.3 in decimal. But the renewable energy project's return is given in thousands of South Korean Won. So, is the 30% referring to a rate, or is it an absolute amount?Looking back at the problem statement: "the return from the renewable energy project equals the maximum rate of return from the community market expansion project." Hmm, the wording is a bit ambiguous. Is it equating the rate of return (which is a percentage) to the amount of return (which is in thousands of Won)?Wait, that doesn't make much sense because one is a percentage and the other is an amount. Maybe I need to interpret it differently. Perhaps the problem is asking when the return from the renewable energy project equals the maximum rate of return in terms of value? Or maybe it's a misstatement, and they actually mean the rate of return for the renewable project equals the maximum rate.But the renewable project's return is given as ( E(t) = 100e^{0.03t} ), which is an amount, not a rate. So, maybe the question is asking when the amount returned by the renewable project equals the maximum rate of return in percentage terms? That would be comparing apples and oranges, so perhaps I'm misunderstanding.Wait, let me re-read the problem statement:"Determine the time ( t ) at which the return from the renewable energy project equals the maximum rate of return from the community market expansion project (from sub-problem 1)."So, the return from the renewable project is ( E(t) ), which is in thousands of Won, and the maximum rate of return is 30%, which is a percentage. So, are they equating 30% to ( E(t) )?But 30% is 0.3 in decimal, but ( E(t) ) is in thousands of Won, so unless we're talking about 30% of some principal amount, but the problem doesn't specify any principal for the renewable project.Wait, hold on. Maybe I need to think of the rate of return for the renewable project. The problem says the return is modeled by ( E(t) = 100e^{0.03t} ). So, is this the total return, or is it the rate?Wait, the wording says "the return from the renewable energy project", so I think ( E(t) ) is the amount returned, not the rate. So, it's in thousands of Won. So, if the maximum rate of return from the community market is 30%, which is a percentage, how can we equate that to an amount?This is confusing. Maybe the problem is expecting me to set ( E(t) = 30 ) because 30% is 0.3, but in thousands of Won, 30% of what? Or maybe they just mean 30,000 Won?Wait, the problem says "the return from the renewable energy project equals the maximum rate of return from the community market expansion project". So, if the maximum rate of return is 30%, which is 0.3 in decimal, but the return from the renewable project is in thousands of Won. So, unless 30% is referring to 30% of the investment, but we don't know the investment amount.Wait, maybe I need to interpret the rate of return as a decimal, so 30% is 0.3, and set ( E(t) = 0.3 ). But ( E(t) ) is in thousands of Won, so 0.3 thousand Won is 300 Won. That seems low, but let's see.Alternatively, maybe the problem is referring to the rate of return for the renewable project. Wait, the renewable project's return is given as ( E(t) = 100e^{0.03t} ). So, is this the amount returned, or is it the rate?Wait, the problem says "the return from the renewable energy project", so it's the amount. So, perhaps the question is asking when the amount returned by the renewable project equals the maximum rate of return in percentage terms, but that would be comparing percentages to amounts, which doesn't make much sense.Alternatively, maybe it's a misstatement, and they actually mean the rate of return for the renewable project equals the maximum rate of return from the community market. But the renewable project's return is given as an exponential function, which is an amount, not a rate.Wait, unless the rate of return for the renewable project is modeled by ( E(t) ), but that seems unlikely because it's given as an exponential function of the amount, not the rate.Wait, maybe I need to think about the rate of return for the renewable project. If ( E(t) = 100e^{0.03t} ), then the rate of return would be the derivative of ( E(t) ) with respect to time, divided by the initial investment or something?Wait, no, the rate of return is usually calculated as (Return - Investment)/Investment. But here, ( E(t) ) is the return, so maybe the rate of return is ( (E(t) - E(0))/E(0) ). Since ( E(0) = 100e^{0} = 100 ). So, the rate of return at time ( t ) would be ( (E(t) - 100)/100 times 100% ), which is ( (E(t)/100 - 1) times 100% ).So, if ( E(t) = 100e^{0.03t} ), then the rate of return ( R_{renewable}(t) ) is ( (100e^{0.03t}/100 - 1) times 100% = (e^{0.03t} - 1) times 100% ).So, the rate of return for the renewable project is ( (e^{0.03t} - 1) times 100% ). So, if we set this equal to the maximum rate of return from the community market, which is 30%, we can solve for ( t ).So, equation:( (e^{0.03t} - 1) times 100% = 30% )Divide both sides by 100%:( e^{0.03t} - 1 = 0.3 )Add 1 to both sides:( e^{0.03t} = 1.3 )Take natural logarithm on both sides:( 0.03t = ln(1.3) )Calculate ( ln(1.3) ). Let me recall that ( ln(1.3) ) is approximately 0.262364.So, ( 0.03t = 0.262364 )Solve for ( t ):( t = 0.262364 / 0.03 )Calculating that: 0.262364 divided by 0.03.Well, 0.262364 / 0.03 is the same as 26.2364 / 3, which is approximately 8.7455 years.So, approximately 8.75 years.Wait, let me verify that.First, ( e^{0.03t} = 1.3 ). Taking natural log:( 0.03t = ln(1.3) approx 0.262364 )So, ( t = 0.262364 / 0.03 approx 8.7455 ) years.Yes, that seems correct.Alternatively, if I didn't consider the rate of return and just set ( E(t) = 30 ), since 30% is 0.3, but in thousands of Won, 30% would be 0.3 thousand, which is 300 Won. But that seems too low because ( E(t) = 100e^{0.03t} ) starts at 100 and grows exponentially. So, 30 is much lower than the starting point. So, that can't be right.Therefore, the correct approach is to equate the rate of return of the renewable project to the maximum rate from the community market.So, the time ( t ) is approximately 8.75 years.But let me express it more precisely. ( ln(1.3) ) is approximately 0.262364264, so:( t = 0.262364264 / 0.03 )Calculating that:0.262364264 divided by 0.03.Well, 0.03 goes into 0.262364264 how many times?0.03 * 8 = 0.240.262364264 - 0.24 = 0.0223642640.03 goes into 0.022364264 approximately 0.745 times.So, total is 8.745 years, which is approximately 8.75 years.So, rounding to two decimal places, 8.75 years.Alternatively, if we want to be more precise, it's 8.7455 years, which is approximately 8.75 years.So, the time ( t ) is approximately 8.75 years.Wait, but let me think again. The problem says "the return from the renewable energy project equals the maximum rate of return from the community market expansion project."If "return" is in thousands of Won, and "rate of return" is a percentage, then equating them directly doesn't make sense. So, perhaps the problem is expecting me to set ( E(t) = 30 ), interpreting 30% as 30 units in the same scale as ( E(t) ), which is thousands of Won.But 30% of what? If the investment amount is not given, it's unclear. Maybe the problem assumes that the rate of return is 30%, so the return amount is 30% of the initial investment. But since the initial investment isn't specified, we can't compute the exact amount.Alternatively, maybe the problem is just asking when ( E(t) = 30 ), interpreting 30% as 30 in the same units as ( E(t) ), which is thousands of Won. So, 30,000 Won.But ( E(t) = 100e^{0.03t} ). So, setting ( 100e^{0.03t} = 30 ).But 30 is less than 100, so ( e^{0.03t} = 0.3 ). Taking natural log:( 0.03t = ln(0.3) approx -1.20397 )So, ( t = -1.20397 / 0.03 approx -40.132 ) years.Negative time doesn't make sense, so that can't be right.Therefore, this approach is invalid.Hence, the correct interpretation is that the rate of return for the renewable project equals the maximum rate of return from the community market, which is 30%. So, we have to compute the rate of return for the renewable project, set it equal to 30%, and solve for ( t ).As I did earlier, leading to ( t approx 8.75 ) years.Therefore, the time ( t ) is approximately 8.75 years.So, summarizing:1. The community market expansion has a maximum rate of return at ( t = 6 ) years, with a rate of 30%.2. The renewable energy project's return equals this maximum rate of return at approximately 8.75 years.I think that's the solution.Final Answer1. The maximum rate of return is achieved at boxed{6} years, with a rate of boxed{30%}.2. The return from the renewable energy project equals the maximum rate of return at approximately boxed{8.75} years.</think>

🧠:<think>Okay, so I have this problem about optimizing backend systems for an e-commerce company. It's about figuring out the maximum number of transactions the system can handle per second without overloading the CPU or I/O subsystems. Then, there's a second part about maintaining stability based on a probability that each transaction completes within a certain time threshold.Let me start with the first part. The system processes N transactions per second. Each transaction takes an average of T milliseconds of CPU time and M milliseconds of I/O time. The CPU can handle C milliseconds per second, and the I/O subsystem can handle I milliseconds per second. I need to model the maximum N without exceeding these capacities.Hmm, so for the CPU, each transaction takes T ms, so for N transactions, the total CPU time needed per second is N*T. Similarly, for I/O, it's N*M. The CPU capacity is C ms/s, so N*T must be less than or equal to C. Similarly, N*M must be less than or equal to I.So, the constraints are:1. N*T ≤ C2. N*M ≤ ITherefore, the maximum N is the minimum of (C / T) and (I / M). So, N_max = min(C / T, I / M). That makes sense because the system is limited by whichever resource (CPU or I/O) is the bottleneck.Wait, but is that all? Are there any other factors? The problem doesn't mention anything else, so I think that's the main model.Now, moving on to the second part. The system must maintain stability, defined by a probability P that any transaction completes within T_max milliseconds. The transaction times follow a normal distribution with mean μ and standard deviation σ. I need to derive an expression for P in terms of μ, σ, and T_max, and then determine the conditions for stability.Okay, so if the transaction times are normally distributed, the probability that a transaction completes within T_max is the probability that a random variable X (transaction time) is less than or equal to T_max. So, P(X ≤ T_max).In terms of the normal distribution, this probability can be found using the cumulative distribution function (CDF). The formula is:P = Φ((T_max - μ) / σ)Where Φ is the CDF of the standard normal distribution.But the problem says to express P in terms of μ, σ, and T_max, so maybe I should write it in terms of the error function or something else? Wait, Φ is typically expressed in terms of the error function. Specifically, Φ(z) = (1/2)(1 + erf(z / sqrt(2))). So, substituting z with (T_max - μ)/σ, we get:P = 0.5 * [1 + erf((T_max - μ) / (σ * sqrt(2)))].But I think the first expression using Φ is sufficient unless they require it in terms of erf. The question says "derive an expression for P", so either form is acceptable, but maybe the standard normal CDF is more straightforward.Now, to determine the conditions under which the system maintains stability, we need to ensure that P is above a certain threshold. Let's say the company wants a high probability, like 99%, that each transaction completes within T_max. So, we can set P ≥ desired probability, say P ≥ 0.99.Using the expression above, we can solve for T_max in terms of μ and σ. Let's denote the desired probability as P_desired. Then,Φ((T_max - μ) / σ) ≥ P_desiredTaking the inverse Φ (the probit function) on both sides:(T_max - μ) / σ ≥ Φ^{-1}(P_desired)Therefore,T_max ≥ μ + σ * Φ^{-1}(P_desired)This gives the condition that T_max must be at least μ plus some multiple of σ, depending on the desired probability. For example, for P = 0.99, Φ^{-1}(0.99) is approximately 2.326, so T_max should be at least μ + 2.326σ.Alternatively, if T_max is fixed, we can solve for the required μ and σ. For instance, if T_max is given, then μ must be less than or equal to T_max - σ * Φ^{-1}(P_desired). This would mean that the mean transaction time must be sufficiently below T_max, considering the variability (σ) and the desired confidence level.So, summarizing, the system maintains stability when T_max is sufficiently large relative to μ and σ, specifically when T_max ≥ μ + σ * Φ^{-1}(P). This ensures that the probability of a transaction completing within T_max is at least P.Wait, but in the first part, we had N_max = min(C / T, I / M). Does this relate to the second part? Maybe not directly, unless the transaction times T and M are related to μ and σ. But in the first part, T and M are average times, so μ would be T + M? Or is μ just the total time per transaction, which is T + M? Hmm, the problem says each transaction requires T ms of CPU and M ms of I/O, so the total time per transaction is T + M. So, μ = T + M.But in the second part, the transaction times are normally distributed with mean μ and standard deviation σ. So, if μ is the average time, which is T + M, then the total time per transaction is variable around that mean with standard deviation σ.But in the first part, we're assuming each transaction takes exactly T and M on average, so maybe in reality, the transaction times are variable, hence the normal distribution in the second part.So, perhaps the first part is a deterministic model, while the second part is a probabilistic model considering variability.Therefore, to ensure stability, we need to set T_max such that the probability P is above a certain threshold, which depends on μ and σ as derived.I think that's the gist of it. Let me just recap:1. For the maximum transactions per second without overloading CPU or I/O, N_max is the minimum of (C / T) and (I / M).2. For stability, the probability P that a transaction completes within T_max is given by the CDF of the normal distribution: P = Φ((T_max - μ)/σ). To maintain stability, T_max must be set such that this probability is above a desired level, which translates to T_max ≥ μ + σ * Φ^{-1}(P).I think that covers both parts. I should make sure I didn't miss any assumptions. The problem states that each transaction requires an average of T ms CPU and M ms I/O, so the total time is T + M, which would be the mean μ. So, μ = T + M. Therefore, in the second part, μ is known as T + M, and σ is given or can be estimated.So, if we have μ = T + M, then the condition becomes T_max ≥ (T + M) + σ * Φ^{-1}(P). This ensures that the probability of a transaction completing within T_max is at least P.Alternatively, if T_max is fixed, then the mean transaction time (T + M) must be less than T_max - σ * Φ^{-1}(P). This could inform how much we can increase T or M before needing to increase T_max or improve σ (reduce variability).I think that's a reasonable approach. I don't see any immediate errors in my reasoning, so I'll proceed to write the final answer based on this.</think>

🧠:<think>Okay, so I have this problem about Mr. Lewis and his hospital room. Let me try to figure it out step by step. First, the room is a rectangular prism, which means it has a length, width, and height. The dimensions are given as 6 meters in length, 4 meters in width, and 3 meters in height. I need to calculate the volume of air that needs to be filtered per hour. Hmm, I remember that the volume of a rectangular prism is calculated by multiplying length, width, and height. So, let me write that down:Volume = length × width × heightPlugging in the numbers:Volume = 6 m × 4 m × 3 mLet me compute that. 6 times 4 is 24, and 24 times 3 is 72. So, the volume is 72 cubic meters. But wait, the problem says the air needs to circulate 5 times per hour. So, does that mean we need to filter the entire volume 5 times each hour? That makes sense because circulating the air 5 times would mean replacing the entire volume five times. So, if the volume is 72 cubic meters, then the amount of air that needs to be filtered per hour is 5 times that volume. Let me write that:Air to be filtered per hour = 5 × VolumeSo, plugging in the numbers:Air to be filtered per hour = 5 × 72 = 360 cubic meters per hour.Okay, that seems straightforward. So, the first part is done. The answer is 360 cubic meters per hour.Now, moving on to the second problem. Mr. Lewis is considering an air filtration system with a filtration rate modeled by the function F(t) = 1000e^{-0.05t} cubic meters per hour. We need to find the time t when the system's filtration rate will no longer meet the required circulation rate of 360 cubic meters per hour.So, essentially, we need to solve for t in the equation:1000e^{-0.05t} = 360Let me write that down:1000e^{-0.05t} = 360I need to solve for t. Hmm, okay, so this is an exponential equation. I think I can use logarithms to solve for t. Let me recall how to do that.First, I can divide both sides by 1000 to isolate the exponential term:e^{-0.05t} = 360 / 1000Calculating 360 divided by 1000, that's 0.36. So,e^{-0.05t} = 0.36Now, to solve for t, I can take the natural logarithm (ln) of both sides. Remember that ln(e^x) = x. So,ln(e^{-0.05t}) = ln(0.36)Simplifying the left side:-0.05t = ln(0.36)Now, I need to compute ln(0.36). Let me recall that ln(1) is 0, and ln of a number less than 1 is negative. So, ln(0.36) will be a negative number.I can use a calculator to find ln(0.36). Let me compute that.Using calculator: ln(0.36) ≈ -1.021651So, plugging that back into the equation:-0.05t = -1.021651Now, to solve for t, I can divide both sides by -0.05:t = (-1.021651) / (-0.05)Dividing two negative numbers gives a positive result. So,t ≈ 1.021651 / 0.05Calculating that:1.021651 divided by 0.05. Hmm, 1 divided by 0.05 is 20, so 1.021651 divided by 0.05 is approximately 20.43302.So, t ≈ 20.433 hours.But, since the problem asks for the time when the filtration rate will no longer meet the required rate, we need to consider that at t ≈ 20.433 hours, the filtration rate drops below 360 cubic meters per hour. Therefore, the system will no longer meet the requirement after approximately 20.433 hours.But, let me double-check my calculations to make sure I didn't make a mistake.Starting from:1000e^{-0.05t} = 360Divide both sides by 1000:e^{-0.05t} = 0.36Take natural log:-0.05t = ln(0.36)Compute ln(0.36):Yes, that's approximately -1.021651.Then, t = (-1.021651)/(-0.05) ≈ 20.433.So, that seems correct.But, just to be thorough, let me plug t = 20.433 back into the original equation to see if it gives approximately 360.F(t) = 1000e^{-0.05*20.433}Compute the exponent:-0.05 * 20.433 ≈ -1.02165So, e^{-1.02165} ≈ e^{-1.02165} ≈ 0.36Therefore, 1000 * 0.36 = 360. That checks out.So, the time t when the filtration rate drops below 360 is approximately 20.433 hours.But, the question says "when the system's filtration rate will no longer meet the required circulation rate." So, that would be at t ≈ 20.433 hours. But, since the problem is about when it will no longer meet, we might need to consider whether to round up or down. Since at t = 20.433, it's exactly 360. So, just after that time, it will drop below. So, depending on the context, sometimes in these problems, you might round to the nearest whole number or keep it as a decimal.But, the question doesn't specify, so probably just give the exact value or round to a reasonable decimal place.Alternatively, maybe express it in terms of exact expression with ln, but since we have a numerical value, 20.433, which is approximately 20.43 hours.But, let me see if I can express it more accurately. Let me compute ln(0.36) more precisely.Using a calculator, ln(0.36):Let me compute it step by step.We know that ln(0.36) is equal to ln(36/100) = ln(9/25) = ln(9) - ln(25) = 2 ln(3) - 2 ln(5)Calculating ln(3) ≈ 1.098612, ln(5) ≈ 1.609438So, 2 ln(3) ≈ 2.1972242 ln(5) ≈ 3.218876So, ln(0.36) = 2.197224 - 3.218876 ≈ -1.021652So, that's consistent with my earlier calculation.Therefore, t ≈ (-1.021652)/(-0.05) ≈ 20.43304So, approximately 20.433 hours.But, maybe we can express it as a fraction or something? Let me see.0.433 hours is approximately how many minutes? 0.433 * 60 ≈ 26 minutes. So, 20 hours and 26 minutes.But, unless the question asks for minutes, probably just leave it in decimal form.Alternatively, if I want to write it as an exact expression, it's t = (ln(0.36))/(-0.05) = (ln(36/100))/(-0.05) = (ln(9/25))/(-0.05) = (ln(9) - ln(25))/(-0.05) = (2 ln 3 - 2 ln 5)/(-0.05) = 2(ln 3 - ln 5)/(-0.05) = (-2)(ln 5 - ln 3)/0.05 = (2)(ln 5 - ln 3)/0.05But, that might be overcomplicating it. Since we have a numerical value, 20.433 hours, that's probably sufficient.So, to recap:1. The volume of the room is 72 cubic meters. Since the air needs to circulate 5 times per hour, the required filtration rate is 5 * 72 = 360 cubic meters per hour.2. The filtration system's rate is given by F(t) = 1000e^{-0.05t}. We set this equal to 360 and solve for t, which gives us approximately 20.433 hours.Therefore, after about 20.43 hours, the filtration rate will drop below the required 360 cubic meters per hour.I think that's it. I don't see any mistakes in my calculations, so I feel confident with these answers.Final Answer1. The volume of air that needs to be filtered per hour is boxed{360} cubic meters.2. The time when the filtration rate will no longer meet the required rate is approximately boxed{20.43} hours.</think>