🧠:<think>Alright, so I've got these two problems to solve about Mr. Smith and his resistance to adopting agile practices. Let me take them one at a time.Problem 1: Calculating the time to double the success with agile practicesOkay, the success function is given by ( S(t) = a cdot e^{kt} ). Without agile, the success rate ( k ) is 0.05, and with agile, it's 0.08. Initially, ( S(0) = 100 ), so that means when ( t = 0 ), ( S(0) = a cdot e^{0} = a cdot 1 = a ). Therefore, ( a = 100 ).So, the success functions are:- Without agile: ( S_{text{no}}(t) = 100 cdot e^{0.05t} )- With agile: ( S_{text{yes}}(t) = 100 cdot e^{0.08t} )We need to find the time ( t ) when ( S_{text{yes}}(t) = 2 cdot S_{text{no}}(t) ).So, setting up the equation:( 100 cdot e^{0.08t} = 2 cdot 100 cdot e^{0.05t} )Simplify both sides by dividing by 100:( e^{0.08t} = 2 cdot e^{0.05t} )Hmm, to solve for ( t ), I can take the natural logarithm of both sides. Let's do that.Taking ln:( ln(e^{0.08t}) = ln(2 cdot e^{0.05t}) )Simplify left side: ( 0.08t )Right side: ( ln(2) + ln(e^{0.05t}) = ln(2) + 0.05t )So now, equation is:( 0.08t = ln(2) + 0.05t )Subtract ( 0.05t ) from both sides:( 0.03t = ln(2) )Therefore, ( t = frac{ln(2)}{0.03} )Calculating that, ( ln(2) ) is approximately 0.6931.So, ( t approx frac{0.6931}{0.03} approx 23.103 ) months.Wait, that seems a bit long. Let me double-check my steps.1. Start with ( S_{text{yes}}(t) = 2 cdot S_{text{no}}(t) )2. Substitute the functions: ( 100e^{0.08t} = 200e^{0.05t} )3. Divide both sides by 100: ( e^{0.08t} = 2e^{0.05t} )4. Divide both sides by ( e^{0.05t} ): ( e^{0.03t} = 2 )5. Take natural log: ( 0.03t = ln(2) )6. So, ( t = ln(2)/0.03 approx 23.103 ) months.Yeah, that seems correct. So, approximately 23.1 months.Problem 2: Deriving the minimum value of ( C ) for positive net benefitAlright, so Mr. Smith has to decide whether to adopt agile practices. The initial cost is 10,000, but the success rate increases as per the model. The monetary value of project success is directly proportional to ( S(t) ) with a proportionality constant ( C ). We need to find the minimum ( C ) such that the net monetary benefit is positive within the first 12 months.First, let's understand what net monetary benefit means here. It should be the total monetary value from the project success minus the initial cost. Since the success is over time, we might need to consider the integral of the success function over the 12 months, but the problem doesn't specify whether it's cumulative or just at the end. Hmm.Wait, the problem says "the monetary value of project success is directly proportional to ( S(t) )". So, does that mean at each time ( t ), the monetary value is ( C cdot S(t) )? Or is it the total over time?I think it's the latter. Because if it's directly proportional, it could mean that the value at each time is ( C cdot S(t) ), but to get the total benefit, we might need to integrate over the period. However, the problem says "net monetary benefit becomes positive in the first 12 months." So, perhaps we need to compute the total benefit over 12 months with agile minus the initial cost, and set that greater than zero.Alternatively, maybe it's the difference in success at 12 months times ( C ) minus the initial cost. Hmm, the wording is a bit ambiguous.Wait, let's read it again: "the monetary value of project success is directly proportional to ( S(t) ) with a proportionality constant ( C ), derive the minimum value of ( C ) such that the net monetary benefit of adopting agile practices becomes positive in the first 12 months."So, perhaps the net benefit is the difference in monetary value between using agile and not using agile, minus the initial cost. So, if we compute the monetary value with agile minus the monetary value without agile, minus the initial cost, and set that greater than zero.But the problem says "the net monetary benefit of adopting agile practices becomes positive". So, it's (monetary value with agile - monetary value without agile - initial cost) > 0.Alternatively, maybe it's (monetary value with agile - initial cost) > (monetary value without agile). Either way, we need to model it.But since the monetary value is directly proportional to ( S(t) ), perhaps we need to compute the integral of ( C cdot S(t) ) over the 12 months for both cases, subtract the initial cost for agile, and set the difference positive.Wait, let's think carefully.If we don't adopt agile, the monetary value over 12 months is ( int_{0}^{12} C cdot S_{text{no}}(t) dt ).If we adopt agile, the monetary value is ( int_{0}^{12} C cdot S_{text{yes}}(t) dt - 10,000 ).So, the net benefit is ( left( int_{0}^{12} C cdot S_{text{yes}}(t) dt - 10,000 right) - int_{0}^{12} C cdot S_{text{no}}(t) dt > 0 ).Simplify that:( int_{0}^{12} C cdot (S_{text{yes}}(t) - S_{text{no}}(t)) dt - 10,000 > 0 )So, ( C cdot int_{0}^{12} (S_{text{yes}}(t) - S_{text{no}}(t)) dt > 10,000 )Therefore, ( C > frac{10,000}{int_{0}^{12} (S_{text{yes}}(t) - S_{text{no}}(t)) dt} )So, we need to compute the integral of ( S_{text{yes}}(t) - S_{text{no}}(t) ) from 0 to 12, then divide 10,000 by that integral to find the minimum ( C ).Let me compute that integral.First, ( S_{text{yes}}(t) = 100e^{0.08t} )( S_{text{no}}(t) = 100e^{0.05t} )So, ( S_{text{yes}}(t) - S_{text{no}}(t) = 100(e^{0.08t} - e^{0.05t}) )Thus, the integral becomes:( int_{0}^{12} 100(e^{0.08t} - e^{0.05t}) dt )Factor out the 100:( 100 int_{0}^{12} (e^{0.08t} - e^{0.05t}) dt )Compute the integral term by term.Integral of ( e^{kt} ) is ( frac{1}{k} e^{kt} ).So,( 100 left[ frac{1}{0.08} e^{0.08t} - frac{1}{0.05} e^{0.05t} right]_{0}^{12} )Compute the antiderivative at 12 and subtract at 0.First, compute at t=12:( frac{1}{0.08} e^{0.08 cdot 12} - frac{1}{0.05} e^{0.05 cdot 12} )Compute exponents:0.08*12 = 0.960.05*12 = 0.60So,( frac{1}{0.08} e^{0.96} - frac{1}{0.05} e^{0.60} )Compute each term:1/0.08 = 12.51/0.05 = 20So,12.5 * e^{0.96} - 20 * e^{0.60}Compute e^{0.96} ≈ e^1 is about 2.718, e^{0.96} is slightly less. Let me calculate:e^{0.96} ≈ 2.6117e^{0.60} ≈ 1.8221So,12.5 * 2.6117 ≈ 32.64620 * 1.8221 ≈ 36.442So, 32.646 - 36.442 ≈ -3.796Wait, that's negative? That can't be right because ( S_{text{yes}}(t) ) is growing faster, so the integral should be positive.Wait, maybe I made a mistake in the signs.Wait, the integral is ( int (e^{0.08t} - e^{0.05t}) dt ), which is positive because 0.08 > 0.05, so ( e^{0.08t} > e^{0.05t} ) for t > 0.But when I computed at t=12, I got 32.646 - 36.442 ≈ -3.796, which is negative. That doesn't make sense.Wait, perhaps I miscalculated the exponents or the multiplication.Wait, let's recalculate e^{0.96} and e^{0.60}.e^{0.96}: Let's compute it more accurately.We know that e^{0.6931} = 2, e^{1} ≈ 2.71828.Compute e^{0.96}:Let me use Taylor series or a calculator approximation.Alternatively, use known values:e^{0.96} ≈ e^{1 - 0.04} = e^1 / e^{0.04} ≈ 2.71828 / 1.04081 ≈ 2.6117 (which matches my previous estimate)Similarly, e^{0.60} ≈ 1.822118800So, 12.5 * 2.6117 ≈ 32.64620 * 1.8221 ≈ 36.442So, 32.646 - 36.442 ≈ -3.796Wait, but that's negative. That can't be, because the integral of a positive function should be positive.Wait, perhaps I messed up the antiderivative.Wait, the integral is:( int (e^{0.08t} - e^{0.05t}) dt = frac{1}{0.08} e^{0.08t} - frac{1}{0.05} e^{0.05t} + C )So, when evaluating from 0 to 12, it's:[ (1/0.08 e^{0.08*12} - 1/0.05 e^{0.05*12}) ] - [ (1/0.08 e^{0} - 1/0.05 e^{0}) ]Which is:[ (12.5 e^{0.96} - 20 e^{0.60}) ] - [ (12.5 * 1 - 20 * 1) ]So, compute the first part:12.5 * 2.6117 ≈ 32.64620 * 1.8221 ≈ 36.442So, 32.646 - 36.442 ≈ -3.796Then subtract the second part:At t=0: 12.5 - 20 = -7.5So, total integral is:(-3.796) - (-7.5) = (-3.796) + 7.5 ≈ 3.704Ah, okay, that makes sense. So, the integral is approximately 3.704.Wait, let me write it step by step.Compute F(t) = (1/0.08)e^{0.08t} - (1/0.05)e^{0.05t}At t=12: F(12) = 12.5 e^{0.96} - 20 e^{0.60} ≈ 12.5*2.6117 - 20*1.8221 ≈ 32.646 - 36.442 ≈ -3.796At t=0: F(0) = 12.5*1 - 20*1 = -7.5So, integral from 0 to 12 is F(12) - F(0) = (-3.796) - (-7.5) = 3.704Therefore, the integral is approximately 3.704.But wait, we had factored out the 100 earlier, so the total integral is 100 * 3.704 ≈ 370.4So, the integral of (S_yes - S_no) from 0 to 12 is approximately 370.4.Therefore, the inequality is:C * 370.4 > 10,000So, C > 10,000 / 370.4 ≈ 26.99So, approximately 27.But let me compute it more accurately.Compute 10,000 / 370.4:370.4 * 27 = 9,999.8, which is almost 10,000.So, 27 * 370.4 = 9,999.8, which is just under 10,000.Therefore, to make it greater than 10,000, C needs to be just over 27.But since we need the minimum C such that the net benefit is positive, we can say C must be greater than approximately 27.But let's compute it more precisely.Compute 10,000 / 370.4:370.4 * 27 = 9,999.8So, 27 gives 9,999.8, which is just 0.2 less than 10,000.So, to get exactly 10,000, we need:C = 10,000 / 370.4 ≈ 26.994So, approximately 26.994, which is roughly 27.00.But since we can't have a fraction of a cent, depending on the context, but since it's a proportionality constant, it can be a decimal.Therefore, the minimum value of C is approximately 26.994, which we can round to 27.00.But let me verify the integral calculation again to ensure accuracy.Compute F(12):12.5 * e^{0.96} ≈ 12.5 * 2.6117 ≈ 32.64620 * e^{0.60} ≈ 20 * 1.8221 ≈ 36.442So, F(12) ≈ 32.646 - 36.442 ≈ -3.796F(0) = 12.5 - 20 = -7.5So, integral is (-3.796) - (-7.5) = 3.704Multiply by 100: 370.4So, 10,000 / 370.4 ≈ 26.994Yes, that's correct.Therefore, the minimum value of C is approximately 26.994, which we can express as 27.00.But to be precise, maybe we should carry more decimal places.Compute 10,000 / 370.4:370.4 * 26.994 ≈ 10,000But let's compute 10,000 / 370.4:370.4 goes into 10,000 how many times?370.4 * 27 = 9,999.8So, 27 gives 9,999.8, which is 0.2 less than 10,000.Therefore, to get exactly 10,000, we need:C = 27 + (0.2 / 370.4) ≈ 27 + 0.00054 ≈ 27.00054So, approximately 27.0005.But since we're dealing with money, usually two decimal places are sufficient, so 27.00 is sufficient.Alternatively, if we need more precision, it's approximately 27.00.But let me check if the integral was correctly calculated.Wait, another way to compute the integral:We have ( int_{0}^{12} (e^{0.08t} - e^{0.05t}) dt )Which is ( left[ frac{1}{0.08} e^{0.08t} - frac{1}{0.05} e^{0.05t} right]_0^{12} )Compute at t=12:12.5 e^{0.96} - 20 e^{0.60} ≈ 12.5*2.6117 - 20*1.8221 ≈ 32.646 - 36.442 ≈ -3.796At t=0:12.5*1 - 20*1 = -7.5So, difference: -3.796 - (-7.5) = 3.704Multiply by 100: 370.4Yes, that's correct.So, 10,000 / 370.4 ≈ 26.994, which is approximately 27.00.Therefore, the minimum value of C is approximately 27.00.But let me think again: is the net benefit (agile - no agile - cost) > 0?Yes, because the net benefit is the additional monetary value from agile minus the initial cost.So, the additional value is C * integral of (S_yes - S_no) from 0 to 12, which is 370.4C.Set 370.4C - 10,000 > 0 => C > 10,000 / 370.4 ≈ 26.994.So, yes, C must be greater than approximately 26.994, so minimum C is approximately 27.00.Alternatively, if we use exact values without approximating e^{0.96} and e^{0.60}, we might get a slightly different result, but for practical purposes, 27 is sufficient.So, summarizing:1. The time required is approximately 23.1 months.2. The minimum C is approximately 27.00.Final Answer1. The time required is boxed{23.1} months.2. The minimum value of ( C ) is boxed{27}.</think>

🧠:<think>Alright, so I have this problem about a web designer optimizing a WordPress site's load time. The response time is given by a function, and I need to find the average over 24 hours. Then, after implementing a caching mechanism, I have to calculate the new average. Hmm, okay, let's take it step by step.First, for part 1, the response time function is ( T(t) = 50 + 10sin(pi t/12) ). I need to find the average over 24 hours. 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 is:[text{Average} = frac{1}{b - a} int_{a}^{b} T(t) , dt]In this case, a is 0 and b is 24, since we're looking at a full day. So, plugging in the values:[text{Average} = frac{1}{24 - 0} int_{0}^{24} left(50 + 10sinleft(frac{pi t}{12}right)right) dt]I can split this integral into two parts:[frac{1}{24} left( int_{0}^{24} 50 , dt + int_{0}^{24} 10sinleft(frac{pi t}{12}right) dt right)]Calculating the first integral is straightforward. The integral of 50 with respect to t is 50t. Evaluated from 0 to 24, that's:[50 times 24 - 50 times 0 = 1200]So, the first part is 1200.Now, the second integral is ( int_{0}^{24} 10sinleft(frac{pi t}{12}right) dt ). Let me make a substitution to solve this. Let me set ( u = frac{pi t}{12} ). Then, ( du = frac{pi}{12} dt ), which means ( dt = frac{12}{pi} du ).Changing the limits of integration: when t = 0, u = 0. When t = 24, u = ( frac{pi times 24}{12} = 2pi ).So, substituting, the integral becomes:[10 times int_{0}^{2pi} sin(u) times frac{12}{pi} du = frac{120}{pi} int_{0}^{2pi} sin(u) du]The integral of sin(u) is -cos(u). Evaluating from 0 to 2π:[frac{120}{pi} left[ -cos(2pi) + cos(0) right] = frac{120}{pi} left[ -1 + 1 right] = frac{120}{pi} times 0 = 0]So, the second integral is 0. That makes sense because the sine function is symmetric over its period, which is 24 hours in this case. So, the positive and negative areas cancel out.Therefore, the average response time is:[frac{1}{24} times (1200 + 0) = frac{1200}{24} = 50 text{ milliseconds}]Okay, so the average is 50 ms. That seems straightforward.Now, moving on to part 2. The caching mechanism reduces the response time by a factor proportional to ( e^{-kt} ). The cache is cleared every 6 hours, and k is 0.1. So, I need to model this effect and find the new average over 24 hours.Hmm, so every 6 hours, the cache is cleared, meaning the reduction factor starts fresh. So, the response time after caching would be ( T(t) times e^{-kt} ), but only for the period since the last cache clear. So, over 24 hours, the cache is cleared 4 times (since 24 / 6 = 4). Each time, the function resets.Therefore, I think I need to model the response time as a piecewise function, where in each 6-hour interval, the response time is ( T(t) times e^{-k(t - n times 6)} ), where n is the number of cache clears that have occurred before time t.So, for t between 0 and 6, it's ( T(t) times e^{-0.1 t} ).For t between 6 and 12, it's ( T(t) times e^{-0.1 (t - 6)} ).Similarly, for t between 12 and 18, it's ( T(t) times e^{-0.1 (t - 12)} ).And for t between 18 and 24, it's ( T(t) times e^{-0.1 (t - 18)} ).Therefore, to find the average, I need to compute the integral over each 6-hour interval, multiply by the respective exponential decay, and then add them all up, then divide by 24.So, the average response time ( overline{T} ) is:[overline{T} = frac{1}{24} left( int_{0}^{6} T(t) e^{-0.1 t} dt + int_{6}^{12} T(t) e^{-0.1 (t - 6)} dt + int_{12}^{18} T(t) e^{-0.1 (t - 12)} dt + int_{18}^{24} T(t) e^{-0.1 (t - 18)} dt right)]Since each integral is similar, just shifted by 6 hours each time, I can compute one integral and then multiply by 4, but I need to be careful because the function T(t) is periodic with period 24 hours, but in each interval, the exponential factor is different.Wait, actually, T(t) is 50 + 10 sin(π t /12). So, in each interval, the sine function is just a shifted version. Let me see.Alternatively, maybe I can compute one integral from 0 to 6, then shift t accordingly for the next intervals.But perhaps it's better to compute each integral separately.Let me first compute the integral from 0 to 6:[I_1 = int_{0}^{6} left(50 + 10sinleft(frac{pi t}{12}right)right) e^{-0.1 t} dt]Similarly, for the next interval, t is from 6 to 12. Let me make a substitution: let u = t - 6. Then, when t = 6, u = 0; when t = 12, u = 6. So, T(t) becomes T(u + 6):[T(u + 6) = 50 + 10sinleft(frac{pi (u + 6)}{12}right) = 50 + 10sinleft(frac{pi u}{12} + frac{pi}{2}right)]Using the sine addition formula:[sinleft(frac{pi u}{12} + frac{pi}{2}right) = sinleft(frac{pi u}{12}right)cosleft(frac{pi}{2}right) + cosleft(frac{pi u}{12}right)sinleft(frac{pi}{2}right) = cosleft(frac{pi u}{12}right)]So, T(u + 6) = 50 + 10 cos(π u /12)Therefore, the integral from 6 to 12 becomes:[I_2 = int_{0}^{6} left(50 + 10cosleft(frac{pi u}{12}right)right) e^{-0.1 u} du]Similarly, for the third interval, t from 12 to 18, let me set u = t - 12:[T(u + 12) = 50 + 10sinleft(frac{pi (u + 12)}{12}right) = 50 + 10sinleft(frac{pi u}{12} + piright) = 50 - 10sinleft(frac{pi u}{12}right)]So, the integral becomes:[I_3 = int_{0}^{6} left(50 - 10sinleft(frac{pi u}{12}right)right) e^{-0.1 u} du]Similarly, for the last interval, t from 18 to 24, set u = t - 18:[T(u + 18) = 50 + 10sinleft(frac{pi (u + 18)}{12}right) = 50 + 10sinleft(frac{pi u}{12} + frac{3pi}{2}right)]Again, using sine addition:[sinleft(frac{pi u}{12} + frac{3pi}{2}right) = sinleft(frac{pi u}{12}right)cosleft(frac{3pi}{2}right) + cosleft(frac{pi u}{12}right)sinleft(frac{3pi}{2}right) = -cosleft(frac{pi u}{12}right)]So, T(u + 18) = 50 - 10 cos(π u /12)Therefore, the integral becomes:[I_4 = int_{0}^{6} left(50 - 10cosleft(frac{pi u}{12}right)right) e^{-0.1 u} du]So, now, I have four integrals:I1: 0 to 6, T(t) e^{-0.1 t}I2: 0 to 6, (50 + 10 cos(π u /12)) e^{-0.1 u}I3: 0 to 6, (50 - 10 sin(π u /12)) e^{-0.1 u}I4: 0 to 6, (50 - 10 cos(π u /12)) e^{-0.1 u}Hmm, interesting. So, if I compute I1, I2, I3, I4, each over 0 to 6, then sum them up and divide by 24, I get the average.But maybe there's a pattern here. Let me see:I1 is integrating (50 + 10 sin(π t /12)) e^{-0.1 t}I2 is integrating (50 + 10 cos(π t /12)) e^{-0.1 t}I3 is integrating (50 - 10 sin(π t /12)) e^{-0.1 t}I4 is integrating (50 - 10 cos(π t /12)) e^{-0.1 t}So, if I add I1 + I2 + I3 + I4, let's see:I1 + I2 + I3 + I4 = [ (50 + 10 sin) + (50 + 10 cos) + (50 - 10 sin) + (50 - 10 cos) ] e^{-0.1 t} integrated over 0 to 6.Simplify the terms inside the brackets:50 + 50 + 50 + 50 = 20010 sin + 10 cos -10 sin -10 cos = 0So, the entire expression simplifies to 200 e^{-0.1 t}Therefore, the sum of the four integrals is:[int_{0}^{6} 200 e^{-0.1 t} dt]Wait, that's a big simplification! So, instead of computing four separate integrals, I can just compute this one integral and multiply by 1 (since it's already summed up). So, the total integral over 24 hours is:[int_{0}^{6} 200 e^{-0.1 t} dt]Wait, no, actually, each I1, I2, I3, I4 is over 0 to 6, so when I add them, it's equivalent to integrating 200 e^{-0.1 t} from 0 to 6. So, the total integral is:200 times the integral of e^{-0.1 t} from 0 to 6.So, let's compute that:First, compute the integral of e^{-0.1 t} dt.The integral is:[int e^{-0.1 t} dt = frac{e^{-0.1 t}}{-0.1} + C = -10 e^{-0.1 t} + C]So, evaluating from 0 to 6:[-10 e^{-0.1 times 6} + 10 e^{0} = -10 e^{-0.6} + 10(1) = 10(1 - e^{-0.6})]Therefore, the total integral is:200 times that:200 * 10 (1 - e^{-0.6}) = 2000 (1 - e^{-0.6})So, the total integral over 24 hours is 2000 (1 - e^{-0.6})Therefore, the average response time is:[overline{T} = frac{2000 (1 - e^{-0.6})}{24}]Simplify that:First, compute 2000 / 24. Let's divide numerator and denominator by 8: 250 / 3 ≈ 83.333...So, approximately 83.333 * (1 - e^{-0.6})Compute e^{-0.6}: e^{-0.6} ≈ 0.5488So, 1 - 0.5488 ≈ 0.4512Multiply by 83.333: 83.333 * 0.4512 ≈ ?Let me compute 83.333 * 0.4512:First, 83 * 0.4512 = ?83 * 0.4 = 33.283 * 0.05 = 4.1583 * 0.0012 = 0.0996Adding up: 33.2 + 4.15 = 37.35; 37.35 + 0.0996 ≈ 37.4496Then, 0.333 * 0.4512 ≈ 0.1504So, total ≈ 37.4496 + 0.1504 ≈ 37.6So, approximately 37.6 milliseconds.Wait, that seems quite low. Let me check my steps again.Wait, hold on. The total integral over 24 hours is 2000 (1 - e^{-0.6}), which is approximately 2000 * 0.4512 ≈ 902.4Then, average is 902.4 / 24 ≈ 37.6 ms.But wait, the original average was 50 ms, and with caching, it's reduced to about 37.6 ms. That seems plausible because caching should reduce the response time.But let me verify the integral computation again.Wait, I had:Total integral = 200 * ∫₀⁶ e^{-0.1 t} dtWhich is 200 * [ -10 e^{-0.1 t} ]₀⁶ = 200 * [ -10 e^{-0.6} + 10 e^{0} ] = 200 * 10 (1 - e^{-0.6}) = 2000 (1 - e^{-0.6})Yes, that's correct.So, 2000 (1 - e^{-0.6}) ≈ 2000 * 0.4512 ≈ 902.4Divide by 24: 902.4 / 24 = 37.6Yes, that's correct.Alternatively, exact value is (2000 / 24)(1 - e^{-0.6}) = (250 / 3)(1 - e^{-0.6})But 250 / 3 is approximately 83.333, as I had.So, 83.333 * (1 - e^{-0.6}) ≈ 83.333 * 0.4512 ≈ 37.6So, the new average response time is approximately 37.6 milliseconds.But let me compute it more accurately.Compute 2000 * (1 - e^{-0.6}) / 24First, compute e^{-0.6}:e^{-0.6} ≈ 0.548811636So, 1 - e^{-0.6} ≈ 0.451188364Multiply by 2000: 2000 * 0.451188364 ≈ 902.376728Divide by 24: 902.376728 / 24 ≈ 37.5990303So, approximately 37.6 ms.Therefore, the new average server response time is approximately 37.6 milliseconds.Wait, but let me think again. Is this the correct approach?Because in each interval, the exponential factor is applied to the response time, but the response time itself is a function that varies sinusoidally. So, by breaking it into intervals and shifting, I accounted for the periodicity of T(t) and the resetting of the exponential decay every 6 hours.Yes, that seems correct. So, the integrals over each 6-hour period, when summed, give the total integral over 24 hours, which is 2000 (1 - e^{-0.6}), leading to an average of approximately 37.6 ms.So, summarizing:1. The average response time over 24 hours is 50 ms.2. After implementing the caching mechanism, the new average response time is approximately 37.6 ms.I think that's the solution.Final Answer1. The average server response time over a 24-hour period is boxed{50} milliseconds.2. The new average server response time after implementing the caching mechanism is approximately boxed{37.6} milliseconds.</think>

🧠:<think>Okay, so I have this problem where I need to help an IT expert set up a secure communication system using RSA encryption for a police department. The problem is divided into two parts: generating RSA keys and then using those keys to encrypt and decrypt a message. Let me try to work through each step carefully.Starting with the first part: generating the RSA keys. The expert has chosen two prime numbers, p = 61 and q = 53. I remember that in RSA, the modulus n is calculated as the product of p and q. So, let me compute that first.n = p * q = 61 * 53. Hmm, let me do the multiplication. 60*53 is 3180, and 1*53 is 53, so adding them together, 3180 + 53 = 3233. So, n is 3233.Next, I need to compute φ(n), Euler's totient function. Since n is the product of two distinct primes, φ(n) = (p - 1)*(q - 1). So, plugging in the values, φ(n) = (61 - 1)*(53 - 1) = 60 * 52. Let me calculate that: 60*50 is 3000, and 60*2 is 120, so 3000 + 120 = 3120. Therefore, φ(n) is 3120.The public exponent e is given as 17. So, the public key is (n, e) = (3233, 17). Now, I need to find the private key d, which is the modular multiplicative inverse of e modulo φ(n). That means I need to find an integer d such that (e * d) ≡ 1 mod φ(n). In other words, 17*d ≡ 1 mod 3120.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 17 and b is 3120. Since 17 and 3120 are coprime (their GCD is 1), there will be a solution for x, which will be our d.Let me set up the algorithm step by step.First, divide 3120 by 17 to find the quotient and remainder.3120 ÷ 17. Let me compute 17*183 = 3111 (since 17*180=3060, and 17*3=51, so 3060+51=3111). Then, 3120 - 3111 = 9. So, the remainder is 9.So, 3120 = 17*183 + 9.Now, take 17 and divide by the remainder 9.17 ÷ 9 = 1 with a remainder of 8. So, 17 = 9*1 + 8.Next, divide 9 by the remainder 8.9 ÷ 8 = 1 with a remainder of 1. So, 9 = 8*1 + 1.Then, divide 8 by the remainder 1.8 ÷ 1 = 8 with a remainder of 0. So, we've reached the GCD, which is 1.Now, working backwards to express 1 as a linear combination of 17 and 3120.Starting from the last non-zero remainder, which is 1:1 = 9 - 8*1But 8 = 17 - 9*1, from the previous step.Substituting that in:1 = 9 - (17 - 9*1)*1 = 9 - 17 + 9 = 2*9 - 17Now, 9 = 3120 - 17*183, from the first step.Substituting that in:1 = 2*(3120 - 17*183) - 17 = 2*3120 - 366*17 - 17 = 2*3120 - 367*17So, 1 = (-367)*17 + 2*3120This means that x = -367 is a solution to 17*x ≡ 1 mod 3120. However, we want a positive value for d, so we add 3120 to -367 until we get a positive number.Let me compute -367 + 3120 = 2753. Let me check if 2753 is positive. Yes, it is. So, d = 2753.Wait, let me verify that 17*2753 mod 3120 is indeed 1.Calculating 17*2753: 17*2000 = 34,000; 17*700 = 11,900; 17*50 = 850; 17*3 = 51. So, adding those together: 34,000 + 11,900 = 45,900; 45,900 + 850 = 46,750; 46,750 + 51 = 46,801.Now, divide 46,801 by 3120 to find the remainder.3120*15 = 46,800. So, 46,801 - 46,800 = 1. Therefore, 17*2753 = 46,801 ≡ 1 mod 3120. Perfect, that checks out.So, the private key d is 2753.Now, moving on to the second part: encrypting and decrypting the message m = 42.First, encryption. The formula for encryption in RSA is c = m^e mod n.So, c = 42^17 mod 3233. Hmm, that's a big exponent. I need a way to compute this efficiently. I can use the method of exponentiation by squaring.Let me break down 42^17 mod 3233.First, compute powers of 42 modulo 3233:Compute 42^1 mod 3233 = 4242^2 = 42*42 = 1764 mod 3233 = 176442^4 = (42^2)^2 = 1764^2. Let me compute 1764*1764. Hmm, that's a bit large. Maybe I can compute it modulo 3233 step by step.Alternatively, I can compute 1764^2 mod 3233.But 1764 is less than 3233, so squaring it would give a number less than 3233^2, which is manageable.Wait, 1764^2 = (1700 + 64)^2 = 1700^2 + 2*1700*64 + 64^2.1700^2 = 2,890,0002*1700*64 = 2*108,800 = 217,60064^2 = 4,096Adding them together: 2,890,000 + 217,600 = 3,107,600; 3,107,600 + 4,096 = 3,111,696.Now, compute 3,111,696 mod 3233.To find this, divide 3,111,696 by 3233 and find the remainder.First, find how many times 3233 fits into 3,111,696.Compute 3233 * 960 = ?Well, 3233 * 1000 = 3,233,000. That's more than 3,111,696.So, let's try 3233 * 960.3233 * 900 = 2,909,7003233 * 60 = 193,980So, 2,909,700 + 193,980 = 3,103,680Subtract that from 3,111,696: 3,111,696 - 3,103,680 = 8,016.Now, compute 8,016 mod 3233.3233 * 2 = 6,4668,016 - 6,466 = 1,550So, 8,016 mod 3233 = 1,550Therefore, 42^4 mod 3233 = 1,550.Wait, that seems a bit high, but let's proceed.Now, 42^8 = (42^4)^2 = 1,550^2 mod 3233.Compute 1,550^2 = 2,402,500Now, divide 2,402,500 by 3233.3233 * 742 = ?Well, 3233 * 700 = 2,263,1003233 * 40 = 129,3203233 * 2 = 6,466Adding them: 2,263,100 + 129,320 = 2,392,420; 2,392,420 + 6,466 = 2,398,886Subtract from 2,402,500: 2,402,500 - 2,398,886 = 3,614Now, 3,614 mod 3233 = 3,614 - 3233 = 381So, 42^8 mod 3233 = 381.Next, 42^16 = (42^8)^2 = 381^2 mod 3233.Compute 381^2 = 145,161Now, divide 145,161 by 3233.3233 * 44 = 142,252145,161 - 142,252 = 2,909So, 42^16 mod 3233 = 2,909.Now, we have 42^16 mod 3233 = 2,909 and we need 42^17.So, 42^17 = 42^16 * 42 mod 3233 = 2,909 * 42 mod 3233.Compute 2,909 * 42.2,909 * 40 = 116,3602,909 * 2 = 5,818Adding them: 116,360 + 5,818 = 122,178Now, compute 122,178 mod 3233.Divide 122,178 by 3233.3233 * 37 = ?3233*30=96,9903233*7=22,631Adding: 96,990 + 22,631 = 119,621Subtract from 122,178: 122,178 - 119,621 = 2,557So, 42^17 mod 3233 = 2,557.Therefore, the ciphertext c is 2,557.Now, let's decrypt this ciphertext using the private key d = 2753.The decryption formula is m = c^d mod n.So, m = 2,557^2753 mod 3233. That's a huge exponent. I need a smart way to compute this.I can use the method of exponentiation by squaring again, but since the exponent is so large, it might be time-consuming. Alternatively, I can use the Chinese Remainder Theorem (CRT) since I know the factors p and q.Wait, the problem doesn't specify whether to use CRT or not, but since I have p and q, maybe it's easier. Let me try that.First, compute c mod p and c mod q.c = 2,557Compute 2,557 mod 61:61*41 = 2,5012,557 - 2,501 = 56So, c mod p = 56Similarly, compute 2,557 mod 53:53*48 = 2,5442,557 - 2,544 = 13So, c mod q = 13Now, compute m_p = (c mod p)^d mod p = 56^2753 mod 61Similarly, compute m_q = (c mod q)^d mod q = 13^2753 mod 53Then, use CRT to find m such that m ≡ m_p mod 61 and m ≡ m_q mod 53.First, compute m_p = 56^2753 mod 61.But since 61 is prime, by Fermat's little theorem, 56^(60) ≡ 1 mod 61. So, we can reduce the exponent 2753 mod 60.Compute 2753 ÷ 60: 60*45 = 2700, so 2753 - 2700 = 53. So, 2753 ≡ 53 mod 60.Therefore, 56^2753 ≡ 56^53 mod 61.Now, compute 56^53 mod 61. Hmm, 56 is congruent to -5 mod 61, so 56 ≡ -5 mod 61.So, (-5)^53 mod 61. Since 53 is odd, this is -5^53 mod 61.Compute 5^53 mod 61.Again, using Fermat's little theorem, 5^60 ≡ 1 mod 61. So, 5^53 = 5^(60 - 7) = (5^60)*(5^-7) ≡ 1*(5^-7) mod 61.So, need to find 5^-7 mod 61, which is the inverse of 5^7 mod 61.First, compute 5^7 mod 61.5^1 = 55^2 = 255^3 = 125 mod 61 = 125 - 2*61 = 125 - 122 = 35^4 = 5^3 * 5 = 3*5 = 155^5 = 15*5 = 75 mod 61 = 75 - 61 = 145^6 = 14*5 = 70 mod 61 = 70 - 61 = 95^7 = 9*5 = 45 mod 61 = 45So, 5^7 ≡ 45 mod 61. Therefore, 5^-7 ≡ 45^-1 mod 61.Find the inverse of 45 mod 61. We need to find x such that 45x ≡ 1 mod 61.Using the Extended Euclidean Algorithm:61 = 1*45 + 1645 = 2*16 + 1316 = 1*13 + 313 = 4*3 + 13 = 3*1 + 0Now, backtracking:1 = 13 - 4*3But 3 = 16 - 1*13, so:1 = 13 - 4*(16 - 13) = 5*13 - 4*16But 13 = 45 - 2*16, so:1 = 5*(45 - 2*16) - 4*16 = 5*45 - 10*16 - 4*16 = 5*45 - 14*16But 16 = 61 - 1*45, so:1 = 5*45 - 14*(61 - 45) = 5*45 - 14*61 + 14*45 = 19*45 - 14*61Therefore, 19*45 ≡ 1 mod 61. So, the inverse of 45 mod 61 is 19.Thus, 5^-7 ≡ 19 mod 61.Therefore, 5^53 ≡ 19 mod 61.So, (-5)^53 ≡ -19 mod 61. Since -19 mod 61 is 42 (because 61 - 19 = 42), we have m_p = 42 mod 61.Now, compute m_q = 13^2753 mod 53.Again, since 53 is prime, by Fermat's little theorem, 13^52 ≡ 1 mod 53. So, reduce the exponent 2753 mod 52.Compute 2753 ÷ 52: 52*52 = 2704, so 2753 - 2704 = 49. So, 2753 ≡ 49 mod 52.Thus, 13^2753 ≡ 13^49 mod 53.Compute 13^49 mod 53. Let's see if we can find a pattern or reduce it further.Alternatively, compute 13^k mod 53 step by step.Compute powers of 13 modulo 53:13^1 = 1313^2 = 169 mod 53. 53*3=159, so 169 - 159 = 10. So, 13^2 ≡ 10 mod 53.13^4 = (13^2)^2 = 10^2 = 100 mod 53. 100 - 53 = 47. So, 13^4 ≡ 47 mod 53.13^8 = (13^4)^2 = 47^2 = 2209 mod 53. Let's compute 53*41=2173, so 2209 - 2173=36. So, 13^8 ≡ 36 mod 53.13^16 = (13^8)^2 = 36^2 = 1296 mod 53. 53*24=1272, so 1296 - 1272=24. So, 13^16 ≡ 24 mod 53.13^32 = (13^16)^2 = 24^2 = 576 mod 53. 53*10=530, 576 - 530=46. So, 13^32 ≡ 46 mod 53.Now, we need 13^49. Let's express 49 as 32 + 16 + 1.So, 13^49 = 13^32 * 13^16 * 13^1 mod 53.We have:13^32 ≡ 4613^16 ≡ 2413^1 ≡ 13So, multiply them together:46 * 24 = 1,1041,104 mod 53: 53*20=1,060, so 1,104 - 1,060=44Then, 44 * 13 = 572572 mod 53: 53*10=530, 572 - 530=42So, 13^49 ≡ 42 mod 53.Therefore, m_q = 42 mod 53.Now, we have:m ≡ 42 mod 61m ≡ 42 mod 53We need to find m such that it satisfies both congruences. Since 42 is less than both 61 and 53, and 42 is congruent to itself mod both, the solution is m = 42.Therefore, the decrypted message is 42, which matches the original message. So, the encryption and decryption worked correctly.Alternatively, if I didn't use CRT, I could have computed 2,557^2753 mod 3233 directly, but that would have been computationally intensive. Using CRT made it much easier.So, summarizing:1. Public key (n, e) = (3233, 17)2. Private key d = 27533. Encrypted message c = 25574. Decrypted message m = 42Everything checks out.Final AnswerThe public key is ( boxed{(3233, 17)} ), the private key is ( boxed{2753} ), the ciphertext is ( boxed{2557} ), and the decrypted message is ( boxed{42} ).</think>

🧠:<think>Okay, so I have this problem about collision detection between two moving objects, and then a second part about optimizing it for multiple objects. Let me try to figure this out step by step.Starting with the first part: two objects moving along parametric lines. Their paths are given by:Object 1:x₁(t) = a₁ + b₁ ty₁(t) = c₁ + d₁ tObject 2:x₂(t) = a₂ + b₂ ty₂(t) = c₂ + d₂ tI need to find the time t when they collide, within [0, T]. So, collision means their x and y coordinates are equal at the same time t.So, set x₁(t) = x₂(t) and y₁(t) = y₂(t).From x₁(t) = x₂(t):a₁ + b₁ t = a₂ + b₂ tSimilarly, from y₁(t) = y₂(t):c₁ + d₁ t = c₂ + d₂ tSo, I can write two equations:1. a₁ - a₂ + (b₁ - b₂) t = 02. c₁ - c₂ + (d₁ - d₂) t = 0These are linear equations in t. Let me denote:Equation 1: (b₁ - b₂) t = a₂ - a₁Equation 2: (d₁ - d₂) t = c₂ - c₁So, solving for t from both equations:t = (a₂ - a₁)/(b₁ - b₂) [from equation 1]t = (c₂ - c₁)/(d₁ - d₂) [from equation 2]For the objects to collide, these two t's must be equal, right? So, the collision time t must satisfy both equations.Therefore, the condition is:(a₂ - a₁)/(b₁ - b₂) = (c₂ - c₁)/(d₁ - d₂)Assuming that b₁ ≠ b₂ and d₁ ≠ d₂, otherwise, the lines might be parallel or coinciding.Wait, but what if b₁ = b₂? Then, equation 1 becomes a₁ = a₂. If a₁ ≠ a₂, then no solution for x, so no collision. If a₁ = a₂, then x coordinates are always equal, so we just need to check the y coordinates.Similarly, if d₁ = d₂, equation 2 becomes c₁ = c₂. If c₁ ≠ c₂, no collision; if c₁ = c₂, then y coordinates are always equal, so we just need to check x coordinates.So, putting it all together:Case 1: b₁ ≠ b₂ and d₁ ≠ d₂Compute t from both equations. If they are equal, then that's the collision time. If not, no collision.Case 2: b₁ = b₂If a₁ ≠ a₂, no collision. If a₁ = a₂, then check if the y's collide. So, set c₁ + d₁ t = c₂ + d₂ t. So, (d₁ - d₂) t = c₂ - c₁. If d₁ ≠ d₂, then t = (c₂ - c₁)/(d₁ - d₂). If d₁ = d₂, then if c₁ = c₂, they are coinciding; otherwise, no collision.Case 3: d₁ = d₂Similarly, if c₁ ≠ c₂, no collision. If c₁ = c₂, check x's. So, (b₁ - b₂) t = a₂ - a₁. If b₁ ≠ b₂, t = (a₂ - a₁)/(b₁ - b₂). If b₁ = b₂, then if a₁ = a₂, coinciding; else, no collision.So, in code terms, I would have to handle these cases.But the question says they do collide within [0, T], so we can assume that such a t exists and is within [0, T].So, the general approach is:1. Check if the lines are parallel in x or y direction.2. If not, solve for t from both x and y equations and see if they match.3. If they do, check if t is within [0, T].So, mathematically, the solution is:If b₁ ≠ b₂ and d₁ ≠ d₂:t_x = (a₂ - a₁)/(b₁ - b₂)t_y = (c₂ - c₁)/(d₁ - d₂)If t_x = t_y, then t = t_x is the collision time.If b₁ = b₂ and a₁ ≠ a₂: no collision.If d₁ = d₂ and c₁ ≠ c₂: no collision.If both b₁ = b₂ and d₁ = d₂:Then, check if a₁ = a₂ and c₁ = c₂. If so, the objects are moving along the same line. So, their paths coincide. So, they are always colliding, but since they are moving, their positions change. Wait, actually, if they are moving along the same line, but with different velocities, they might collide at some point.Wait, no. If both x and y velocities are same, then if their initial positions are same, they are coinciding. If initial positions are different, they are moving in the same direction but never meet.Wait, actually, if b₁ = b₂ and d₁ = d₂, then the velocity vectors are same. So, if their initial positions are same, they are always together. If initial positions are different, they are moving in the same direction with same speed, so they never meet.So, in that case, if a₁ ≠ a₂ or c₁ ≠ c₂, no collision.So, putting it all together, the steps are:1. Check if the velocity vectors are same (b₁ = b₂ and d₁ = d₂). If yes: a. If initial positions are same (a₁ = a₂ and c₁ = c₂), then they are always together, so collision at all times. But since we need a specific t in [0, T], maybe t=0? Or any t? But the problem says they do collide, so perhaps t=0 is the start. b. If initial positions are different, no collision.2. If velocity vectors are not same: a. If b₁ ≠ b₂ and d₁ ≠ d₂, compute t_x and t_y. If t_x = t_y and t is within [0, T], then collision at t. b. If b₁ = b₂: i. If a₁ ≠ a₂, no collision. ii. If a₁ = a₂, solve for t from y equation. If t is within [0, T], collision at t. c. If d₁ = d₂: i. If c₁ ≠ c₂, no collision. ii. If c₁ = c₂, solve for t from x equation. If t is within [0, T], collision at t.So, that's the method.Now, for the second part: generalizing to n objects, each with similar parametric equations. Need to find all pairwise collision times within [0, T].So, for n objects, there are C(n, 2) pairs. For each pair, perform the above check.But the computational complexity would be O(n²), since for each pair, we do a constant amount of work.But wait, in practice, each pair's check involves solving two linear equations, which is O(1) per pair. So, overall, it's O(n²) time.But if n is large, say 10^4, then n² is 10^8, which is manageable, but for n=10^5, it's 10^10, which is too big.But in the context of video games, n is probably not that large. So, O(n²) is acceptable.But maybe there are optimizations.Wait, the grandparent was working in the early era, so maybe n wasn't too big, so O(n²) was acceptable. But for modern games with thousands of objects, O(n²) is too slow.But the question is about deriving a general method and discussing the computational complexity.So, the method is:For each pair of objects (i, j), where i < j:1. Check if their paths intersect within [0, T]. a. Compute t as above. b. If t is within [0, T], record the collision time.So, the algorithm is:Initialize an empty list of collision times.For i from 0 to n-1: For j from i+1 to n-1: Compute collision time t between object i and object j. If t is within [0, T], add t to the list.Return the list.The computational complexity is O(n²), since for each pair, we do O(1) operations.But in reality, each pair's check involves solving two linear equations, which is a few arithmetic operations, so it's manageable for small n.However, for large n, this is not efficient. So, optimizations could be done, like spatial partitioning (e.g., grid-based or quadtree) to reduce the number of pairs checked. But the question is about the general method, so O(n²) is the answer.So, summarizing:1. For two objects, solve for t by equating x and y coordinates, handle edge cases where velocities are same or lines are parallel.2. For n objects, check all pairs, resulting in O(n²) complexity.</think>