🧠:<think>Okay, so I have this problem about Alex studying PTSD symptoms over time. There are two parts. Let me try to tackle them one by one.Starting with the first part: The function given is ( P(t) = 4 + 2e^{-0.5t} cos(pi t) ). I need to find the rate of change at ( t = 3 ) months. That means I have to find the derivative of ( P(t) ) with respect to ( t ) and then plug in ( t = 3 ).Alright, so let me recall how to differentiate functions like this. The function has an exponential part and a cosine part multiplied together, so I'll need to use the product rule. The product rule says that if you have ( u(t) cdot v(t) ), the derivative is ( u'(t)v(t) + u(t)v'(t) ).Let me define ( u(t) = 2e^{-0.5t} ) and ( v(t) = cos(pi t) ). So, first, I need to find ( u'(t) ) and ( v'(t) ).Starting with ( u(t) = 2e^{-0.5t} ). The derivative of ( e^{kt} ) is ( ke^{kt} ), so here, ( k = -0.5 ). Therefore, ( u'(t) = 2 times (-0.5)e^{-0.5t} = -e^{-0.5t} ).Next, ( v(t) = cos(pi t) ). The derivative of ( cos(kt) ) is ( -k sin(kt) ). So, ( v'(t) = -pi sin(pi t) ).Now, applying the product rule: ( frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t) ).Plugging in the derivatives:( frac{d}{dt}[2e^{-0.5t} cos(pi t)] = (-e^{-0.5t}) cos(pi t) + 2e^{-0.5t} (-pi sin(pi t)) ).Simplify that:( -e^{-0.5t} cos(pi t) - 2pi e^{-0.5t} sin(pi t) ).So, the derivative of ( P(t) ) is:( P'(t) = 0 + ) [the derivative of the second term] because the first term is 4, whose derivative is 0. So,( P'(t) = -e^{-0.5t} cos(pi t) - 2pi e^{-0.5t} sin(pi t) ).Now, I need to evaluate this at ( t = 3 ).First, let me compute each part step by step.Compute ( e^{-0.5 times 3} = e^{-1.5} ). I remember that ( e^{-1} ) is approximately 0.3679, so ( e^{-1.5} ) is about 0.2231.Next, compute ( cos(pi times 3) ). Since ( cos(3pi) ) is equal to ( cos(pi) ) because cosine has a period of ( 2pi ). ( cos(pi) = -1 ).Then, compute ( sin(pi times 3) ). Similarly, ( sin(3pi) = sin(pi) = 0 ).So, plugging these into the derivative:( P'(3) = -e^{-1.5} times (-1) - 2pi e^{-1.5} times 0 ).Simplify:First term: ( -e^{-1.5} times (-1) = e^{-1.5} approx 0.2231 ).Second term: ( -2pi e^{-1.5} times 0 = 0 ).So, ( P'(3) approx 0.2231 ).Wait, that seems positive. So, the rate of change is positive at ( t = 3 ). That means the PTSD symptoms are increasing at that moment.But let me double-check my calculations because sometimes signs can be tricky.Looking back at the derivative:( P'(t) = -e^{-0.5t} cos(pi t) - 2pi e^{-0.5t} sin(pi t) ).At ( t = 3 ):( cos(3pi) = -1 ), so the first term is ( -e^{-1.5} times (-1) = e^{-1.5} ).( sin(3pi) = 0 ), so the second term is 0.So, yes, ( P'(3) = e^{-1.5} approx 0.2231 ). So, the rate of change is approximately 0.2231 per month.Interpreting this, it means that at 3 months after exposure, the PTSD symptom severity is increasing at a rate of about 0.22 points per month on the scale from 0 to 10.Wait, but the function is ( P(t) = 4 + 2e^{-0.5t} cos(pi t) ). Let me just check if I differentiated correctly.Yes, the derivative of 4 is 0, and then the derivative of the second term is as I computed. So, I think that's correct.Moving on to the second part: Alex studies the effect of therapy starting at ( t = 3 ). The therapy effect is modeled by ( T(t) = e^{-0.2(t - 3)} ) for ( t geq 3 ). So, the new function is ( P_T(t) = P(t) times T(t) ) for ( t geq 3 ).So, ( P_T(t) = [4 + 2e^{-0.5t} cos(pi t)] times e^{-0.2(t - 3)} ).We need to calculate the PTSD score at ( t = 6 ) months under this therapy.So, first, let me write ( P_T(6) = P(6) times T(6) ).Compute ( P(6) ) first.( P(6) = 4 + 2e^{-0.5 times 6} cos(pi times 6) ).Compute each part:( e^{-0.5 times 6} = e^{-3} approx 0.0498 ).( cos(6pi) = cos(0) = 1 ) because cosine has a period of ( 2pi ), so ( 6pi ) is 3 full periods, back to 0.So, ( P(6) = 4 + 2 times 0.0498 times 1 = 4 + 0.0996 approx 4.0996 ).Now, compute ( T(6) = e^{-0.2(6 - 3)} = e^{-0.2 times 3} = e^{-0.6} approx 0.5488 ).Therefore, ( P_T(6) = 4.0996 times 0.5488 approx ).Let me compute that:4.0996 * 0.5488.First, 4 * 0.5488 = 2.1952.0.0996 * 0.5488 ≈ 0.0547.So, total ≈ 2.1952 + 0.0547 ≈ 2.2499.So, approximately 2.25.Wait, that seems a bit low. Let me check my calculations again.Wait, ( P(6) ) is 4 + 2e^{-3} cos(6π). Since cos(6π) is 1, so 2e^{-3} is about 2*0.0498 ≈ 0.0996. So, 4 + 0.0996 ≈ 4.0996. That's correct.Then, ( T(6) = e^{-0.2*(6-3)} = e^{-0.6} ≈ 0.5488 ). Correct.Multiplying 4.0996 * 0.5488:Let me compute 4 * 0.5488 = 2.1952.0.0996 * 0.5488:0.0996 * 0.5 = 0.0498.0.0996 * 0.0488 ≈ approximately 0.00487.So, total ≈ 0.0498 + 0.00487 ≈ 0.05467.Adding to 2.1952: 2.1952 + 0.05467 ≈ 2.24987, which is approximately 2.25.So, yes, about 2.25.But wait, the original P(t) at t=6 is about 4.1, and multiplying by ~0.55 gives ~2.25. That seems correct.Alternatively, maybe I can compute it more accurately.Compute 4.0996 * 0.5488:Let me do 4 * 0.5488 = 2.1952.0.0996 * 0.5488:Compute 0.0996 * 0.5 = 0.0498.0.0996 * 0.0488:Compute 0.0996 * 0.04 = 0.003984.0.0996 * 0.0088 ≈ 0.000876.So, total ≈ 0.003984 + 0.000876 ≈ 0.00486.So, 0.0498 + 0.00486 ≈ 0.05466.Thus, total P_T(6) ≈ 2.1952 + 0.05466 ≈ 2.24986, which is ~2.25.So, approximately 2.25.Alternatively, maybe I can use more precise values.Compute e^{-3} ≈ 0.049787.So, 2 * 0.049787 ≈ 0.099574.Thus, P(6) = 4 + 0.099574 ≈ 4.099574.T(6) = e^{-0.6} ≈ 0.5488116.So, P_T(6) = 4.099574 * 0.5488116.Let me compute this more accurately:4.099574 * 0.5488116.Break it down:4 * 0.5488116 = 2.1952464.0.099574 * 0.5488116.Compute 0.099574 * 0.5 = 0.049787.0.099574 * 0.0488116 ≈ let's compute 0.099574 * 0.04 = 0.00398296.0.099574 * 0.0088116 ≈ approximately 0.000878.So, total ≈ 0.00398296 + 0.000878 ≈ 0.004861.Thus, 0.049787 + 0.004861 ≈ 0.054648.Adding to 2.1952464: 2.1952464 + 0.054648 ≈ 2.2498944.So, approximately 2.25.Therefore, the PTSD score at t=6 under therapy is approximately 2.25.Wait, but the original P(t) at t=6 is about 4.1, and after therapy, it's about 2.25. That's a significant reduction, which makes sense because the therapy effect is multiplicative and exponential decay.So, summarizing:1. The rate of change at t=3 is approximately 0.2231, meaning symptoms are increasing at that point.2. The PTSD score at t=6 under therapy is approximately 2.25.Now, interpreting both results in the context of Alex's study.For part 1, the positive rate of change at 3 months suggests that, without therapy, PTSD symptoms are worsening at that time. This could indicate a period where symptoms are becoming more severe, possibly due to the nature of the trauma and the body's response over time.For part 2, the application of therapy starting at t=3 leads to a significant reduction in symptoms by t=6. The score drops from about 4.1 to 2.25, which is a notable improvement. This suggests that the therapy is effective in reducing PTSD symptoms over time, especially when applied continuously after the initial exposure.So, Alex's findings show that while PTSD symptoms may worsen around the 3-month mark, effective therapy can mitigate this progression and lead to a reduction in symptom severity by 6 months.I think that's a reasonable interpretation. Let me just make sure I didn't make any calculation errors.Double-checking the derivative:Yes, the derivative was computed correctly with the product rule, and the values at t=3 were correctly substituted.For the second part, the multiplication of P(t) and T(t) seems correct, and the calculations at t=6 were done step by step, leading to approximately 2.25.So, I think I'm confident with these results.Final Answer1. The rate of change at ( t = 3 ) months is (boxed{0.223}) (approximately), indicating an increase in PTSD symptoms.2. The PTSD symptom severity score at ( t = 6 ) months under therapy is (boxed{2.25}).</think>

🧠:<think>Okay, so I have this problem about a data analyst assessing the impact of cloud migration on two data centers, A and B. The problem has two parts. Let me try to work through them step by step.First, part 1 is about calculating the new average processing times after the cloud migration. Before migration, Data Center A had an average processing time of 120 minutes, and Data Center B had 150 minutes. After migration, Data Center A's processing time was reduced by 25%, and Data Center B's by 30%. Alright, so to find the new average processing times, I need to calculate 25% of 120 minutes and subtract that from 120 for Data Center A. Similarly, for Data Center B, I need to find 30% of 150 minutes and subtract that from 150.Let me write that down:For Data Center A:25% of 120 minutes is 0.25 * 120 = 30 minutes.So, the new average processing time is 120 - 30 = 90 minutes.For Data Center B:30% of 150 minutes is 0.30 * 150 = 45 minutes.So, the new average processing time is 150 - 45 = 105 minutes.Wait, that seems straightforward. So, after the migration, Data Center A now processes batches in 90 minutes on average, and Data Center B in 105 minutes on average.Okay, moving on to part 2. The data analyst notices that the processing times follow a normal distribution. For Data Center A, the standard deviation post-migration is 15 minutes, and for Data Center B, it's 20 minutes. The question is: If a batch of transactions is chosen at random, what is the probability that the processing time will be less than 100 minutes in Data Center A and less than 120 minutes in Data Center B? We need to use the cumulative distribution function (CDF) of the normal distribution to find these probabilities.Alright, so for each data center, we have a normal distribution with a known mean and standard deviation. We need to find the probability that a randomly selected batch has a processing time less than a certain value. Starting with Data Center A: Mean (μ) is 90 minutes, standard deviation (σ) is 15 minutes. We need P(X < 100).Similarly, for Data Center B: Mean (μ) is 105 minutes, standard deviation (σ) is 20 minutes. We need P(Y < 120).Since these are independent events (assuming the processing times in each data center are independent), the combined probability would be the product of the two individual probabilities.But first, let's compute each probability separately.For Data Center A:We need to find P(X < 100) where X ~ N(90, 15²). To use the CDF, we can standardize the value by calculating the z-score.Z = (X - μ) / σ = (100 - 90) / 15 = 10 / 15 ≈ 0.6667.So, we need the probability that Z < 0.6667. Looking up this z-score in the standard normal distribution table or using a calculator, we can find the CDF value.Similarly, for Data Center B:We need to find P(Y < 120) where Y ~ N(105, 20²). Again, calculate the z-score.Z = (120 - 105) / 20 = 15 / 20 = 0.75.So, we need the probability that Z < 0.75.Now, let me recall the standard normal distribution table values.For Z = 0.6667, which is approximately 0.67. Looking at the z-table, the value for 0.67 is about 0.7486. Alternatively, using a calculator or a more precise method, the exact value can be found.Similarly, for Z = 0.75, the value is approximately 0.7734.But wait, let me confirm these values because sometimes different tables or calculators might have slightly different approximations.Alternatively, using the formula for the CDF of the normal distribution:Φ(z) = (1/2) [1 + erf(z / sqrt(2))]Where erf is the error function. But since I don't have a calculator here, I'll rely on standard z-table values.For Z = 0.6667 (which is 2/3), the CDF is approximately 0.7486.For Z = 0.75, the CDF is approximately 0.7734.So, for Data Center A, P(X < 100) ≈ 0.7486, and for Data Center B, P(Y < 120) ≈ 0.7734.Since these are independent, the combined probability is 0.7486 * 0.7734.Let me compute that:0.7486 * 0.7734 ≈ ?First, multiply 0.7 * 0.7 = 0.49Then, 0.7 * 0.0734 ≈ 0.05138Then, 0.0486 * 0.7 ≈ 0.03402And 0.0486 * 0.0734 ≈ ~0.00357Adding all together: 0.49 + 0.05138 + 0.03402 + 0.00357 ≈ 0.57897Wait, that seems a bit rough. Maybe a better way is to compute 0.7486 * 0.7734.Let me compute 7486 * 7734 first, then adjust the decimal.But that might be time-consuming. Alternatively, approximate:0.7486 is approximately 0.75, and 0.7734 is approximately 0.77.0.75 * 0.77 = 0.5775.So, approximately 0.5775 or 57.75%.But let me compute it more accurately.0.7486 * 0.7734:First, 0.7 * 0.7 = 0.490.7 * 0.0734 = 0.051380.0486 * 0.7 = 0.034020.0486 * 0.0734 ≈ 0.00357Adding all these:0.49 + 0.05138 = 0.541380.54138 + 0.03402 = 0.57540.5754 + 0.00357 ≈ 0.57897So, approximately 0.579 or 57.9%.So, the probability that a randomly chosen batch has processing time less than 100 minutes in Data Center A and less than 120 minutes in Data Center B is approximately 57.9%.Wait, but let me double-check the z-scores and the corresponding probabilities because sometimes the z-table might have different decimal places.For Z = 0.6667, which is 2/3, the exact value can be found using a calculator or precise z-table.Looking it up, Z = 0.6667 corresponds to approximately 0.7486 as I had before.For Z = 0.75, it's 0.7734.Multiplying 0.7486 * 0.7734:Let me compute 0.7486 * 0.7734:First, 0.7 * 0.7 = 0.490.7 * 0.0734 = 0.051380.0486 * 0.7 = 0.034020.0486 * 0.0734 ≈ 0.00357Adding these: 0.49 + 0.05138 = 0.541380.54138 + 0.03402 = 0.57540.5754 + 0.00357 ≈ 0.57897So, approximately 0.579 or 57.9%.Alternatively, using a calculator for more precision:0.7486 * 0.7734 = ?Let me compute 7486 * 7734:But that's too time-consuming. Alternatively, use the fact that 0.7486 * 0.7734 ≈ (0.75 - 0.0014) * (0.77 + 0.0034)Using the formula (a - b)(c + d) = ac + ad - bc - bd= 0.75*0.77 + 0.75*0.0034 - 0.0014*0.77 - 0.0014*0.0034= 0.5775 + 0.00255 - 0.001078 - 0.00000476= 0.5775 + 0.00255 = 0.580.58 - 0.001078 = 0.5789220.578922 - 0.00000476 ≈ 0.578917So, approximately 0.5789 or 57.89%.So, about 57.9%.Therefore, the probability is approximately 57.9%.But let me make sure that I didn't make a mistake in calculating the z-scores.For Data Center A:X = 100, μ = 90, σ = 15.Z = (100 - 90)/15 = 10/15 = 2/3 ≈ 0.6667. Correct.For Data Center B:Y = 120, μ = 105, σ = 20.Z = (120 - 105)/20 = 15/20 = 0.75. Correct.So, the z-scores are correct.Now, the CDF values:For Z = 0.6667, Φ(0.6667) ≈ 0.7486.For Z = 0.75, Φ(0.75) ≈ 0.7734.Multiplying these gives approximately 0.5789 or 57.89%.So, rounding to two decimal places, 57.9%.Alternatively, if we want to express it as a percentage, it's approximately 57.9%.Therefore, the probability is approximately 57.9%.But let me check if the question asks for the probability that both events happen, i.e., processing time less than 100 in A AND less than 120 in B. Since the processing times in each data center are independent, we can multiply the probabilities.Yes, that's correct.So, summarizing:1. New average processing times:Data Center A: 90 minutes.Data Center B: 105 minutes.2. Probability that processing time is less than 100 in A and less than 120 in B is approximately 57.9%.But let me express this as a decimal or a fraction if needed, but the question says to use the CDF, so we can present it as a decimal or percentage.Alternatively, if we want to be more precise, perhaps using a calculator for the exact CDF values.Using a calculator or software:For Z = 0.6667, Φ(Z) ≈ 0.74857.For Z = 0.75, Φ(Z) ≈ 0.77337.Multiplying these: 0.74857 * 0.77337 ≈ ?Let me compute this more accurately.0.74857 * 0.77337:First, multiply 0.7 * 0.7 = 0.490.7 * 0.07337 = 0.0513590.04857 * 0.7 = 0.0339990.04857 * 0.07337 ≈ 0.00357Adding these:0.49 + 0.051359 = 0.5413590.541359 + 0.033999 = 0.5753580.575358 + 0.00357 ≈ 0.578928So, approximately 0.5789 or 57.89%.So, about 57.9%.Therefore, the final answer is approximately 57.9%.But let me check if the question wants the answer in a specific format, like a fraction or a decimal. Since it's a probability, decimal is fine, but sometimes they prefer fractions or percentages.But the question says to use the CDF, so I think decimal is acceptable.Alternatively, if we use more precise z-table values, perhaps the result is slightly different, but I think 0.579 is a reasonable approximation.So, to recap:1. New average processing times:Data Center A: 90 minutes.Data Center B: 105 minutes.2. Probability: Approximately 57.9%.But let me write the exact decimal value as 0.5789, which is approximately 0.579.So, the probability is approximately 0.579 or 57.9%.I think that's it.</think>

🧠:<think>Alright, so I have this problem with two sub-problems. Let me tackle them one by one. Starting with Sub-problem 1: I need to find the points of intersection of the migration paths of Family A and Family B. Both families have parametric equations for their x and y coordinates over time t, which ranges from 0 to 50. Family A's path is given by:- ( x_A(t) = 2t + 3 )- ( y_A(t) = t^2 - t + 1 )Family B's path is given by:- ( x_B(t) = 3t + 1 )- ( y_B(t) = 2t^2 - 3t + 4 )To find the points of intersection, I need to find the values of t where both x and y coordinates are equal for both families. That means solving the system of equations:1. ( 2t + 3 = 3s + 1 ) (since x_A(t) = x_B(s))2. ( t^2 - t + 1 = 2s^2 - 3s + 4 ) (since y_A(t) = y_B(s))Wait, hold on. I just realized that the parameter t is the same for both families, right? Because both families are moving over the same time period. So actually, I should set their x(t) equal and y(t) equal at the same t. So, it's not s and t, but both equations should be equal at the same t. That makes more sense.So, correct equations to solve are:1. ( 2t + 3 = 3t + 1 )2. ( t^2 - t + 1 = 2t^2 - 3t + 4 )Let me solve equation 1 first:( 2t + 3 = 3t + 1 )Subtract 2t from both sides:( 3 = t + 1 )Subtract 1:( t = 2 )Okay, so t is 2. Now, let's check if this t satisfies equation 2.Plug t = 2 into y_A(t):( y_A(2) = (2)^2 - 2 + 1 = 4 - 2 + 1 = 3 )Now, plug t = 2 into y_B(t):( y_B(2) = 2*(2)^2 - 3*(2) + 4 = 8 - 6 + 4 = 6 )Wait, that's not equal. y_A(2) is 3 and y_B(2) is 6. So, that means even though x_A(t) and x_B(t) are equal at t=2, their y-coordinates aren't. Therefore, they don't intersect at t=2.Hmm, that's confusing. Maybe I made a mistake in setting up the equations. Let me think again.Wait, perhaps I need to find t and s such that x_A(t) = x_B(s) and y_A(t) = y_B(s). Because maybe the families are moving at different rates, so their paths could cross at different times. So, t and s don't have to be the same.So, let's denote t for Family A and s for Family B. Then, we have:1. ( 2t + 3 = 3s + 1 )2. ( t^2 - t + 1 = 2s^2 - 3s + 4 )Now, we have two equations with two variables, t and s. Let's solve equation 1 for t or s.From equation 1:( 2t + 3 = 3s + 1 )Let's solve for t:( 2t = 3s + 1 - 3 )( 2t = 3s - 2 )( t = (3s - 2)/2 )Now, substitute this expression for t into equation 2:( [(3s - 2)/2]^2 - [(3s - 2)/2] + 1 = 2s^2 - 3s + 4 )Let me compute each term step by step.First, compute [(3s - 2)/2]^2:= (9s² - 12s + 4)/4Then, compute [(3s - 2)/2]:= (3s - 2)/2So, putting it all together:(9s² - 12s + 4)/4 - (3s - 2)/2 + 1 = 2s² - 3s + 4Let me multiply every term by 4 to eliminate denominators:(9s² - 12s + 4) - 2*(3s - 2) + 4 = 4*(2s² - 3s + 4)Simplify each part:Left side:9s² - 12s + 4 - 6s + 4 + 4Combine like terms:9s² - 18s + 12Right side:8s² - 12s + 16Now, bring all terms to left side:9s² - 18s + 12 - 8s² + 12s - 16 = 0Simplify:(9s² - 8s²) + (-18s + 12s) + (12 - 16) = 0= s² - 6s - 4 = 0So, quadratic equation: s² - 6s - 4 = 0Solve for s using quadratic formula:s = [6 ± sqrt(36 + 16)] / 2= [6 ± sqrt(52)] / 2= [6 ± 2*sqrt(13)] / 2= 3 ± sqrt(13)So, s ≈ 3 + 3.6055 ≈ 6.6055 or s ≈ 3 - 3.6055 ≈ -0.6055Since s represents time, it can't be negative. So, s ≈ 6.6055Now, find t from earlier equation:t = (3s - 2)/2Plug s ≈ 6.6055:t ≈ (3*6.6055 - 2)/2≈ (19.8165 - 2)/2≈ 17.8165/2≈ 8.90825So, t ≈ 8.90825 and s ≈ 6.6055Now, check if these t and s are within the 0 to 50 range. Yes, both are.Now, let's find the coordinates.Compute x_A(t) = 2t + 3 ≈ 2*8.90825 + 3 ≈ 17.8165 + 3 ≈ 20.8165Similarly, x_B(s) = 3s + 1 ≈ 3*6.6055 + 1 ≈ 19.8165 + 1 ≈ 20.8165Good, x coordinates match.Now, compute y_A(t) = t² - t + 1 ≈ (8.90825)^2 - 8.90825 + 1Calculate (8.90825)^2:≈ 79.36So, y_A ≈ 79.36 - 8.90825 + 1 ≈ 71.45175Similarly, y_B(s) = 2s² - 3s + 4 ≈ 2*(6.6055)^2 - 3*6.6055 + 4Calculate (6.6055)^2 ≈ 43.63So, 2*43.63 ≈ 87.26Then, 3*6.6055 ≈ 19.8165Thus, y_B ≈ 87.26 - 19.8165 + 4 ≈ 71.4435Which is approximately equal to y_A(t). The slight difference is due to rounding errors.Therefore, the paths intersect at approximately t ≈ 8.908 and s ≈ 6.605, with coordinates approximately (20.8165, 71.45).Wait, but the problem says "within the 50-year period," so t and s are both within 0 to 50, so this is a valid intersection.Is there another intersection? Let's see.We had s ≈ 6.6055 and s ≈ -0.6055. The negative s is invalid, so only one intersection point.Thus, the paths intersect once at approximately (20.8165, 71.45) at t ≈ 8.908 for Family A and s ≈ 6.605 for Family B.But the question says "points of intersection," so maybe it's just one point.Wait, but let me double-check my calculations.Wait, when I solved the quadratic equation, I got s = 3 ± sqrt(13). sqrt(13) is approximately 3.6055, so s ≈ 6.6055 and s ≈ -0.6055. So, only one valid s.Therefore, only one point of intersection.So, the coordinates are approximately (20.8165, 71.45). But to be precise, maybe I can write exact expressions.From earlier, s = 3 + sqrt(13), so t = (3*(3 + sqrt(13)) - 2)/2 = (9 + 3sqrt(13) - 2)/2 = (7 + 3sqrt(13))/2Similarly, x_A(t) = 2t + 3 = 2*(7 + 3sqrt(13))/2 + 3 = (7 + 3sqrt(13)) + 3 = 10 + 3sqrt(13)Similarly, y_A(t) = t² - t + 1. Let me compute t²:t = (7 + 3sqrt(13))/2t² = [(7)^2 + 2*7*3sqrt(13) + (3sqrt(13))^2]/4= [49 + 42sqrt(13) + 9*13]/4= [49 + 42sqrt(13) + 117]/4= [166 + 42sqrt(13)]/4Then, y_A(t) = t² - t + 1= [166 + 42sqrt(13)]/4 - (7 + 3sqrt(13))/2 + 1Convert all terms to quarters:= [166 + 42sqrt(13)]/4 - [14 + 6sqrt(13)]/4 + 4/4= [166 + 42sqrt(13) - 14 - 6sqrt(13) + 4]/4= [156 + 36sqrt(13)]/4= [39 + 9sqrt(13)]/1= 39 + 9sqrt(13)Wait, that can't be right because earlier approximate calculation was around 71.45, but 39 + 9*3.6055 ≈ 39 + 32.45 ≈ 71.45, which matches. So, exact y-coordinate is 39 + 9sqrt(13).Similarly, x-coordinate is 10 + 3sqrt(13).So, the exact point is (10 + 3sqrt(13), 39 + 9sqrt(13)).Alternatively, we can factor:x = 10 + 3sqrt(13)y = 39 + 9sqrt(13) = 3*(13) + 9sqrt(13) = 3*(13 + 3sqrt(13))But maybe it's better to leave it as is.So, the point of intersection is at (10 + 3√13, 39 + 9√13).Now, moving to Sub-problem 2:Dr. Martin wants to model the density of settlers in a region over time. The region is a 100x100 grid, and the number of families in each cell at time t follows a Poisson distribution with mean λ(t) = α e^{-β t}, where α=5 and β=0.1.We need to calculate the expected number of families in a grid cell at t=25 and t=50.Since it's a Poisson distribution, the expected value is equal to the mean λ(t).So, for t=25:λ(25) = 5 * e^{-0.1*25} = 5 * e^{-2.5}Similarly, for t=50:λ(50) = 5 * e^{-0.1*50} = 5 * e^{-5}Compute these values.First, e^{-2.5} ≈ e^{-2} * e^{-0.5} ≈ 0.1353 * 0.6065 ≈ 0.0821So, λ(25) ≈ 5 * 0.0821 ≈ 0.4105Similarly, e^{-5} ≈ 0.006737947So, λ(50) ≈ 5 * 0.006737947 ≈ 0.03369Therefore, the expected number of families at t=25 is approximately 0.4105, and at t=50 is approximately 0.0337.But since the question asks for the expected number, which is λ(t), so we can present it as exact expressions or approximate decimals.Alternatively, we can write them as:At t=25: 5 e^{-2.5}At t=50: 5 e^{-5}But since the question says "calculate," it's better to provide numerical values.So, rounding to, say, four decimal places:At t=25: ≈ 0.4105At t=50: ≈ 0.0337Alternatively, if more precision is needed, but I think four decimals are sufficient.So, summarizing:Sub-problem 1: The paths intersect at one point: (10 + 3√13, 39 + 9√13) approximately (20.816, 71.45).Sub-problem 2: Expected number of families at t=25 is approximately 0.4105, and at t=50 is approximately 0.0337.</think>

🧠:<think>First, I recognize that the number of books checked out each hour follows a geometric sequence. The first term of this sequence is 5 books, and by the third hour, the number of books checked out is 45.To find the common ratio, I'll use the formula for the nth term of a geometric sequence: ( a_n = a_1 times r^{(n-1)} ). Plugging in the known values, I have ( 45 = 5 times r^{2} ). Solving for ( r ), I find that ( r = 3 ).Now that I have the common ratio, I can calculate the number of books checked out each hour up to the fifth hour. The sequence will be:- 1st hour: 5 books- 2nd hour: 15 books- 3rd hour: 45 books- 4th hour: 135 books- 5th hour: 405 booksAdding these together, the total number of books checked out by the end of the fifth hour is 605.Next, I need to address the inventory balance. The total books checked out are 100, and the current ratio of checked out to returned books is 3:1. This means 75 books are checked out and 25 are returned.To achieve a balance of 2:1, I'll set up the equation ( frac{75}{25 + x} = 2 ), where ( x ) represents the additional books that need to be returned. Solving for ( x ), I find that 12.5 additional books must be returned. Since the number of books must be a whole number, I'll round up to 13 books.</think>