🧠:<think>Alright, so Jenna is a horse owner who's really worried about the side effects of conventional veterinary medicines. She wants to figure out which treatment is safer for her horse between Treatment A and Treatment B. I need to help her by calculating the probabilities of severe side effects for both treatments, considering both the direct severe effects and the escalation from minor to severe. Let me break this down step by step.First, let's tackle the first sub-problem: calculating the overall probability of severe side effects from Treatment A. From the information given, Treatment A has a 15% chance of minor side effects and a 5% chance of severe side effects. Now, if a horse experiences minor side effects, there's a 60% chance these could escalate to severe unless treated with additional medication, which has an 80% success rate.Hmm, okay. So, the overall probability of severe side effects from Treatment A isn't just the direct 5%, but also includes the cases where minor side effects escalate. So, I need to calculate the probability of minor side effects leading to severe ones and then add that to the direct severe side effect probability.Let me write this out:1. Direct severe side effects: 5% or 0.05.2. Minor side effects: 15% or 0.15.3. Probability that minor side effects escalate without treatment: 60% or 0.6.4. Success rate of the additional medication: 80% or 0.8. So, the probability that the escalation is prevented is 0.8, meaning the probability that it isn't prevented is 1 - 0.8 = 0.2.Wait, so if the minor side effects occur, there's a 60% chance they'll escalate, but Jenna can use the additional medication which has an 80% success rate. So, does that mean the probability of escalation is 60% times 20% (since 80% is prevented)? Let me think.Yes, that makes sense. So, the probability that minor side effects lead to severe side effects is the probability of minor side effects times the probability that they escalate despite the medication. So, it's 0.15 (minor) * 0.6 (escalation chance) * (1 - 0.8) (failure of medication to prevent escalation). Wait, no, hold on. Is the 60% chance of escalation already considering the treatment, or is it the chance without treatment?Looking back at the problem statement: "if a horse experiences minor side effects from either treatment, there is a 60% chance that these side effects will escalate to severe side effects unless treated with an additional medication, which has an 80% success rate of preventing the escalation."So, the 60% is the chance of escalation without treatment. The additional medication has an 80% success rate of preventing it. So, the probability of escalation with treatment is 60% * (1 - 0.8) = 60% * 20% = 12%.Therefore, the probability that minor side effects lead to severe side effects is 0.15 * 0.6 * 0.2.Wait, no, hold on. Let me parse this again. The 60% is the chance of escalation unless treated. So, if treated, the success rate is 80%, meaning the failure rate is 20%. So, the probability of escalation is 60% * 20% = 12%. So, the total probability from minor to severe is 15% * 12%? Wait, no.Wait, no, that's not quite right. Let me think in terms of probabilities.If minor side effects occur (15%), then there's a 60% chance they escalate unless treated. The treatment has an 80% success rate, so the chance of escalation is 60% * (1 - 0.8) = 60% * 20% = 12%. So, the probability of minor leading to severe is 15% * 12%? Wait, no, that would be 0.15 * 0.12 = 0.018, which is 1.8%. That seems low.Wait, maybe I'm overcomplicating. Let's model it step by step.1. The probability of minor side effects is 15%.2. Given minor side effects, the probability of escalation without treatment is 60%.3. But Jenna is using the additional medication, which has an 80% success rate. So, the probability that the escalation is prevented is 80%, so the probability that it isn't prevented is 20%.Therefore, the probability that minor side effects lead to severe side effects is 15% * 20% = 3%.Wait, that seems too simplistic. Alternatively, is it 15% * 60% * 20%? Because it's 15% chance of minor, 60% chance of escalation, and 20% chance that the medication fails.Yes, that's correct. So, it's 0.15 * 0.6 * 0.2 = 0.018 or 1.8%.Therefore, the total probability of severe side effects from Treatment A is the direct 5% plus the 1.8% from minor escalation, which is 6.8%.Wait, let me verify. So, 15% minor, 60% escalation chance, 80% prevention success. So, the chance of minor leading to severe is 15% * (60% * 20%) = 15% * 12% = 1.8%. Then, adding the direct 5%, total severe is 6.8%.Yes, that seems right.Now, moving on to Treatment B. The second sub-problem is to compare Treatment A and Treatment B.Treatment B has a 10% chance of minor side effects and an 8% chance of severe side effects. So, similar to Treatment A, we need to calculate the overall severe side effect probability.So, for Treatment B:1. Direct severe side effects: 8% or 0.08.2. Minor side effects: 10% or 0.10.3. Probability of escalation without treatment: 60% or 0.6.4. Success rate of additional medication: 80% or 0.8, so failure rate is 0.2.Therefore, the probability of minor leading to severe is 10% * 60% * 20% = 0.10 * 0.6 * 0.2 = 0.012 or 1.2%.Adding the direct severe side effects: 8% + 1.2% = 9.2%.Therefore, comparing Treatment A (6.8%) and Treatment B (9.2%), Treatment A has a lower overall probability of severe side effects.Wait, but let me double-check my calculations because 6.8% vs. 9.2% seems a significant difference, but I want to make sure I didn't make a mistake.For Treatment A:- Direct severe: 5%- Minor: 15%- Escalation chance: 60%- Medication failure: 20%- So, minor leading to severe: 15% * 60% * 20% = 1.8%- Total severe: 5% + 1.8% = 6.8%For Treatment B:- Direct severe: 8%- Minor: 10%- Escalation chance: 60%- Medication failure: 20%- So, minor leading to severe: 10% * 60% * 20% = 1.2%- Total severe: 8% + 1.2% = 9.2%Yes, that seems correct. So, Treatment A is statistically safer with a lower overall severe side effect probability.Wait, but hold on. Is the 60% escalation chance independent of the treatment? Or is it a given that Jenna will use the additional medication? The problem says "if a horse experiences minor side effects from either treatment, there is a 60% chance that these side effects will escalate to severe side effects unless treated with an additional medication, which has an 80% success rate of preventing the escalation."So, Jenna is using the additional medication, so the 60% is the chance without treatment, but with treatment, it's reduced by 80%. So, yes, the calculation is correct.Alternatively, if the 60% was already considering the treatment, but the problem states it's unless treated, so without treatment it's 60%, with treatment, it's 60% * 20% = 12% chance of escalation.Wait, no, that's not quite. Let me clarify.The 60% is the chance of escalation without treatment. With treatment, the success rate is 80%, so the chance of escalation is 60% * (1 - 0.8) = 12%. So, the probability of minor leading to severe is 12%.But wait, that 12% is conditional on minor side effects. So, for Treatment A, it's 15% * 12% = 1.8%, and for Treatment B, it's 10% * 12% = 1.2%.Wait, hold on. Is the 12% the probability of escalation given minor side effects and treatment? So, yes, for Treatment A, minor is 15%, and given minor, the chance of severe is 12%, so 15% * 12% = 1.8%. Similarly for Treatment B, 10% * 12% = 1.2%.Therefore, the total severe for A is 5% + 1.8% = 6.8%, and for B, 8% + 1.2% = 9.2%.Yes, that seems correct.So, in conclusion, Treatment A has a lower overall probability of severe side effects (6.8%) compared to Treatment B (9.2%), making Treatment A statistically safer.</think>

🧠:<think>Okay, so I have this problem where a historian is studying a military officer's career across three stages of a conflict: early, middle, and late. The officer's strategic decisions in each stage are modeled by three different functions: E(t), M(t), and L(t). The first part asks for the total number of strategic decisions made from the start of the conflict to t = 3 years. The second part wants the rate of change of the total at t = 3 years. Let me break this down step by step.First, understanding the functions:- E(t) = 5t² + 2t + 8. This is for the early stage. I assume this is valid from t = 0 to some point, maybe t = a, but the problem doesn't specify when each stage ends. Hmm, actually, the problem says "from the start of the conflict to time t = 3 years." So, I think each function is applicable for the entire duration, but perhaps each stage is a different period within the 3 years? Wait, no, the functions are defined for each stage, but without specific time intervals. Wait, maybe each function is defined for each stage, but the stages are sequential. So, the early stage is from t = 0 to t = a, middle from t = a to t = b, and late from t = b to t = 3. But the problem doesn't specify when each stage ends. Hmm, that complicates things. Wait, let me read the problem again. It says, "the number of strategic decisions made by the officer in each period and their respective impacts on the broader geopolitical context." So, E(t) is for the early stage, M(t) for the middle, and L(t) for the late. But without knowing when each stage starts and ends, how can we compute the total from t = 0 to t = 3?Wait, maybe each function is meant to be integrated over their respective stages, but since we don't know the duration of each stage, perhaps the functions are meant to be summed up at each time t? That is, the total number of decisions at time t is E(t) + M(t) + L(t). But that doesn't make much sense because each function is specific to a stage. Alternatively, maybe each function is applicable for the entire duration, but E(t) is dominant in the early stage, M(t) in the middle, and L(t) in the late. But without knowing the exact time divisions, I can't integrate each function over their respective intervals.Wait, perhaps the problem is simpler. Maybe each function is meant to be evaluated at t = 3, and then summed up? But that would give the total at t = 3, not from t = 0 to t = 3. Wait, no, the first question is about the total number of strategic decisions made from the start (t = 0) to t = 3. So, perhaps we need to integrate each function over their respective stages, but since the stages are early, middle, and late, but without knowing when they switch, maybe we have to assume that each function is applicable for the entire duration? That seems odd because each function is specific to a stage.Wait, maybe the functions E(t), M(t), and L(t) are all valid for the entire duration, but each represents the rate of decisions in their respective stages. Hmm, that could be. So, the total number of decisions would be the sum of E(t), M(t), and L(t) integrated from t = 0 to t = 3. But that might not make sense because each function is specific to a stage, not a rate.Wait, perhaps E(t) is the number of decisions in the early stage at time t, M(t) in the middle, and L(t) in the late. So, if we consider that the officer was in the early stage from t = 0 to t = a, middle from t = a to t = b, and late from t = b to t = 3. But since we don't know a and b, maybe the functions are meant to be evaluated at t = 3 for each stage? That doesn't seem right either.Wait, maybe the functions are meant to be cumulative. So, E(t) is the total number of decisions in the early stage up to time t, M(t) is the total in the middle up to time t, and L(t) is the total in the late up to time t. Then, the total decisions from t = 0 to t = 3 would be E(3) + M(3) + L(3). But that seems plausible.Wait, but let me think again. If E(t) is the number of decisions in the early stage at time t, then to get the total from t = 0 to t = 3, we would need to integrate E(t) over that period. Similarly for M(t) and L(t). But without knowing the duration of each stage, we can't do that.Wait, perhaps the functions are meant to be evaluated at t = 3 for each stage, assuming that each stage is 1 year? So, early stage from t = 0 to t = 1, middle from t = 1 to t = 2, and late from t = 2 to t = 3. Then, we can compute E(1), M(1), L(1), but that seems arbitrary.Wait, the problem doesn't specify the duration of each stage, so maybe it's intended that each function is evaluated at t = 3 and summed up? That is, the total number of decisions is E(3) + M(3) + L(3). But that would be the total at t = 3, not the cumulative from t = 0 to t = 3.Wait, perhaps the functions are actually rates, so E(t) is the rate of decisions in the early stage, M(t) in the middle, and L(t) in the late. Then, to find the total number of decisions from t = 0 to t = 3, we need to integrate each function over their respective stages. But again, without knowing when the stages switch, we can't do that.Wait, maybe the functions are meant to be summed up and integrated from t = 0 to t = 3. So, total decisions = ∫₀³ [E(t) + M(t) + L(t)] dt. That would make sense if the officer was making decisions in all stages simultaneously, but that doesn't align with the idea of early, middle, and late stages.Hmm, this is confusing. Let me try to think differently. Maybe each function is applicable for the entire duration, but each represents a different aspect. For example, E(t) is the number of decisions in the early phase up to time t, M(t) is the number in the middle up to time t, and L(t) is the number in the late up to time t. So, the total number of decisions up to time t is E(t) + M(t) + L(t). Therefore, to find the total from t = 0 to t = 3, we just evaluate E(3) + M(3) + L(3). That seems plausible.But wait, let me check the units. E(t) is a function of t, so it's number of decisions as a function of time. So, if we evaluate E(3), that's the number of decisions in the early stage at t = 3. Similarly, M(3) is the number in the middle at t = 3, and L(3) is the number in the late at t = 3. But that would mean the total at t = 3 is E(3) + M(3) + L(3). But the question is about the total from the start to t = 3. So, maybe it's the sum of E(t) + M(t) + L(t) integrated from t = 0 to t = 3? That would give the total number of decisions over the entire period.Wait, that makes more sense. Because if E(t), M(t), and L(t) are rates (number per year), then integrating each from 0 to 3 would give the total number of decisions in each stage, and summing them up would give the overall total. But the problem states E(t), M(t), and L(t) are the number of decisions, not the rate. So, maybe they are cumulative functions. So, E(t) is the total number of decisions in the early stage up to time t, M(t) is the total in the middle up to time t, and L(t) is the total in the late up to time t. Therefore, the total number of decisions from t = 0 to t = 3 is E(3) + M(3) + L(3).But wait, if that's the case, then the functions E(t), M(t), and L(t) are cumulative, so E(3) would be the total decisions in the early stage up to t = 3, which might not make sense because the early stage is only a part of the conflict. So, perhaps the functions are meant to represent the rate of decisions in each stage, and we need to integrate them over their respective time intervals.But without knowing the duration of each stage, we can't do that. Hmm, maybe the problem assumes that each stage is of equal duration, so each stage is 1 year. So, early stage from t = 0 to t = 1, middle from t = 1 to t = 2, and late from t = 2 to t = 3. Then, we can compute the total decisions in each stage by integrating E(t) from 0 to 1, M(t) from 1 to 2, and L(t) from 2 to 3, and sum them up.That seems like a possible approach. Let me try that.So, first, define the stages:- Early stage: t = 0 to t = 1- Middle stage: t = 1 to t = 2- Late stage: t = 2 to t = 3Then, the total decisions would be:∫₀¹ E(t) dt + ∫₁² M(t) dt + ∫₂³ L(t) dtThat makes sense because each function is applicable to its respective stage, and we integrate over the duration of each stage.Okay, so let's proceed with that assumption.First, compute ∫₀¹ E(t) dt, where E(t) = 5t² + 2t + 8.Integrate term by term:∫(5t²) dt = (5/3)t³∫(2t) dt = t²∫8 dt = 8tSo, the integral from 0 to 1 is:[(5/3)(1)³ + (1)² + 8(1)] - [(5/3)(0)³ + (0)² + 8(0)] = (5/3 + 1 + 8) - 0 = (5/3 + 9) = (5/3 + 27/3) = 32/3 ≈ 10.6667Next, compute ∫₁² M(t) dt, where M(t) = 3t³ - 4t² + 7t + 5.Integrate term by term:∫(3t³) dt = (3/4)t⁴∫(-4t²) dt = (-4/3)t³∫(7t) dt = (7/2)t²∫5 dt = 5tSo, the integral from 1 to 2 is:[(3/4)(2)⁴ + (-4/3)(2)³ + (7/2)(2)² + 5(2)] - [(3/4)(1)⁴ + (-4/3)(1)³ + (7/2)(1)² + 5(1)]Compute each part:At t = 2:(3/4)(16) = 12(-4/3)(8) = -32/3 ≈ -10.6667(7/2)(4) = 145(2) = 10Total: 12 - 10.6667 + 14 + 10 = (12 + 14 + 10) - 10.6667 = 36 - 10.6667 = 25.3333At t = 1:(3/4)(1) = 0.75(-4/3)(1) = -1.3333(7/2)(1) = 3.55(1) = 5Total: 0.75 - 1.3333 + 3.5 + 5 = (0.75 + 3.5 + 5) - 1.3333 = 9.25 - 1.3333 ≈ 7.9167So, the integral from 1 to 2 is 25.3333 - 7.9167 ≈ 17.4166Now, compute ∫₂³ L(t) dt, where L(t) = 6e^(0.5t) + 10.Integrate term by term:∫6e^(0.5t) dt = 6 * (2)e^(0.5t) = 12e^(0.5t)∫10 dt = 10tSo, the integral from 2 to 3 is:[12e^(0.5*3) + 10*3] - [12e^(0.5*2) + 10*2]Compute each part:At t = 3:12e^(1.5) ≈ 12 * 4.4817 ≈ 53.780410*3 = 30Total: 53.7804 + 30 = 83.7804At t = 2:12e^(1) ≈ 12 * 2.7183 ≈ 32.619610*2 = 20Total: 32.6196 + 20 = 52.6196So, the integral from 2 to 3 is 83.7804 - 52.6196 ≈ 31.1608Now, sum up all three integrals:Early stage: ≈10.6667Middle stage: ≈17.4166Late stage: ≈31.1608Total ≈10.6667 + 17.4166 + 31.1608 ≈59.2431So, the total number of strategic decisions made from t = 0 to t = 3 is approximately 59.24.But let me check if this approach is correct. The problem didn't specify the duration of each stage, so assuming each stage is 1 year might not be accurate. Maybe the stages are longer or shorter. But since the total time is 3 years, and there are three stages, it's logical to assume each stage is 1 year. Otherwise, without more information, we can't determine the exact total.Alternatively, if the functions E(t), M(t), and L(t) are meant to be applied over the entire 3-year period, but each represents a different aspect, then the total would be the sum of their integrals from 0 to 3. But that would be:∫₀³ [E(t) + M(t) + L(t)] dtBut that would mean the officer is making decisions in all stages simultaneously, which doesn't align with the idea of early, middle, and late stages.Therefore, the initial approach of dividing the 3 years into three 1-year stages and integrating each function over its respective interval seems the most reasonable given the lack of specific stage durations.Now, moving on to the second part: Calculate the rate of change of the total number of strategic decisions made at t = 3 years.The rate of change would be the derivative of the total number of decisions with respect to time. Since the total is the sum of the integrals of E(t), M(t), and L(t) over their respective stages, the rate of change at t = 3 would be the derivative of the total function at that point.But wait, the total function is a piecewise function, where each stage contributes to the total. So, up to t = 1, the total is the integral of E(t). From t = 1 to t = 2, it's the integral of E(t) plus the integral of M(t). From t = 2 to t = 3, it's the integral of E(t) plus the integral of M(t) plus the integral of L(t).Therefore, the total number of decisions T(t) is:For t in [0,1]: T(t) = ∫₀ᵗ E(s) dsFor t in [1,2]: T(t) = ∫₀¹ E(s) ds + ∫₁ᵗ M(s) dsFor t in [2,3]: T(t) = ∫₀¹ E(s) ds + ∫₁² M(s) ds + ∫₂ᵗ L(s) dsTherefore, the derivative of T(t) with respect to t is:For t in [0,1]: dT/dt = E(t)For t in [1,2]: dT/dt = M(t)For t in [2,3]: dT/dt = L(t)Therefore, at t = 3, which is in the late stage, the rate of change is L(t) evaluated at t = 3.So, compute L(3):L(t) = 6e^(0.5t) + 10L(3) = 6e^(1.5) + 10 ≈6 * 4.4817 + 10 ≈26.8902 + 10 ≈36.8902So, the rate of change at t = 3 is approximately 36.89 decisions per year.Wait, but let me confirm. Since t = 3 is the end of the late stage, the rate of change is indeed L(t) at t = 3, because up to t = 3, the total is the sum of all three integrals, and the derivative at t = 3 is just L(3), as the other stages have already been integrated up to their respective end points.Yes, that makes sense.So, summarizing:1. Total decisions from t = 0 to t = 3: approximately 59.242. Rate of change at t = 3: approximately 36.89But let me compute these more accurately without rounding too much.First, for the total:Early stage integral:∫₀¹ (5t² + 2t + 8) dt = [ (5/3)t³ + t² + 8t ] from 0 to 1At t = 1: 5/3 + 1 + 8 = 5/3 + 9 = 32/3 ≈10.6666667Middle stage integral:∫₁² (3t³ -4t² +7t +5) dt = [ (3/4)t⁴ - (4/3)t³ + (7/2)t² +5t ] from 1 to 2At t = 2:(3/4)(16) = 12(-4/3)(8) = -32/3 ≈-10.6666667(7/2)(4) =145*2=10Total: 12 - 10.6666667 +14 +10 = 12 +14 +10 =36; 36 -10.6666667=25.3333333At t =1:(3/4)(1)=0.75(-4/3)(1)= -1.3333333(7/2)(1)=3.55*1=5Total:0.75 -1.3333333 +3.5 +5=0.75 +3.5=4.25; 4.25 +5=9.25; 9.25 -1.3333333≈7.9166667So, integral from 1 to2:25.3333333 -7.9166667≈17.4166666Late stage integral:∫₂³ (6e^{0.5t} +10) dt = [12e^{0.5t} +10t] from 2 to3At t=3:12e^{1.5} +30e^{1.5}≈4.4816890712*4.48168907≈53.780268853.7802688 +30=83.7802688At t=2:12e^{1} +20e≈2.71828182812*2.718281828≈32.619381932.6193819 +20=52.6193819Integral from2 to3:83.7802688 -52.6193819≈31.1608869Total decisions:10.6666667 +17.4166666 +31.1608869≈59.2432192So, approximately 59.24.For the rate of change at t=3, it's L(t)=6e^{0.5t} +10At t=3:6e^{1.5} +10≈6*4.48168907 +10≈26.8901344 +10≈36.8901344So, approximately 36.89.But let me express these more precisely, maybe in fractions or exact terms.For the total:Early stage integral:32/3Middle stage integral: Let's compute it exactly.∫₁² (3t³ -4t² +7t +5) dt = [ (3/4)t⁴ - (4/3)t³ + (7/2)t² +5t ] from1 to2At t=2:(3/4)(16)=12(-4/3)(8)= -32/3(7/2)(4)=145*2=10Total:12 -32/3 +14 +10Convert to thirds:12=36/314=42/310=30/3So, 36/3 -32/3 +42/3 +30/3= (36 -32 +42 +30)/3=76/3≈25.3333333At t=1:(3/4)(1)=3/4(-4/3)(1)= -4/3(7/2)(1)=7/25*1=5Total:3/4 -4/3 +7/2 +5Convert to twelfths:3/4=9/12-4/3=-16/127/2=42/125=60/12Total:9/12 -16/12 +42/12 +60/12= (9 -16 +42 +60)/12=95/12≈7.9166667So, integral from1 to2:76/3 -95/12= (76*4 -95)/12=(304 -95)/12=209/12≈17.4166667Late stage integral:∫₂³ (6e^{0.5t} +10) dt= [12e^{0.5t} +10t] from2 to3At t=3:12e^{1.5} +30At t=2:12e^{1} +20So, the integral is12(e^{1.5} -e) +10(3 -2)=12(e^{1.5} -e) +10We can leave it in terms of e if needed, but since the problem asks for a numerical value, we can compute it as:12*(4.48168907 -2.718281828) +10≈12*(1.76340724) +10≈21.1608869 +10≈31.1608869So, total decisions:32/3 +209/12 +31.1608869Convert 32/3 to twelfths:128/12209/12 is already twelfthsSo, 128/12 +209/12=337/12≈28.0833333Then, 28.0833333 +31.1608869≈59.2442202So, approximately 59.24.For the rate of change at t=3, it's L(3)=6e^{1.5} +10≈6*4.48168907 +10≈26.8901344 +10≈36.8901344≈36.89So, rounding to two decimal places, 36.89.Alternatively, if we want to express it more precisely, we can write it as 6e^{1.5} +10, but the problem likely expects a numerical value.Therefore, the answers are approximately 59.24 and 36.89.But let me check if the problem expects exact expressions or decimal approximations.The problem says "determine the total number" and "calculate the rate of change," and it mentions that e≈2.71828, so it's expecting numerical answers.Therefore, the total number is approximately 59.24, and the rate of change is approximately 36.89.But to be precise, let me compute the integrals more accurately.First, early stage:∫₀¹ (5t² +2t +8) dt= [ (5/3)t³ +t² +8t ] from0 to1=5/3 +1 +8=5/3 +9=32/3≈10.6666667Middle stage:∫₁² (3t³ -4t² +7t +5) dt= [ (3/4)t⁴ - (4/3)t³ + (7/2)t² +5t ] from1 to2At t=2:3/4*(16)=12-4/3*(8)= -32/3≈-10.66666677/2*(4)=145*2=10Total:12 -10.6666667 +14 +10=12+14+10=36; 36 -10.6666667=25.3333333At t=1:3/4=0.75-4/3≈-1.33333337/2=3.55=5Total:0.75 -1.3333333 +3.5 +5=0.75+3.5=4.25; 4.25+5=9.25; 9.25 -1.3333333≈7.9166667So, integral from1 to2=25.3333333 -7.9166667≈17.4166666Late stage:∫₂³ (6e^{0.5t} +10) dt= [12e^{0.5t} +10t] from2 to3At t=3:12e^{1.5} +30At t=2:12e^{1} +20Compute e^{1.5} and e:e≈2.718281828e^{1.5}=e^(3/2)=sqrt(e^3)=sqrt(20.0855369232)≈4.48168907So, 12*4.48168907≈53.780268853.7802688 +30=83.7802688At t=2:12*e≈12*2.718281828≈32.619381932.6193819 +20=52.6193819Integral from2 to3=83.7802688 -52.6193819≈31.1608869Total decisions:10.6666667 +17.4166666 +31.1608869≈59.2432192So, approximately 59.24.Rate of change at t=3:L(3)=6e^{1.5} +10≈6*4.48168907 +10≈26.8901344 +10≈36.8901344≈36.89Therefore, the answers are approximately 59.24 and 36.89.But to be precise, let me carry out the calculations with more decimal places.Compute e^{1.5}:e≈2.718281828459045e^{1.5}=e^(3/2)=sqrt(e^3)=sqrt(20.0855369232)=4.4816890703380645So, 6*e^{1.5}=6*4.4816890703380645≈26.890134422028387Adding 10:26.890134422028387 +10=36.890134422028387≈36.8901344So, rate of change≈36.8901344≈36.89Similarly, total decisions:Early stage:32/3≈10.666666666666666Middle stage:209/12≈17.416666666666668Late stage:31.160886899999996Sum:10.666666666666666 +17.416666666666668≈28.08333333333333228.083333333333332 +31.160886899999996≈59.24422023333333So, total≈59.24422023333333≈59.24Therefore, the answers are approximately 59.24 and 36.89.I think that's as precise as we can get without more decimal places.</think>

🧠:<think>Okay, so I'm trying to help Alex optimize their carbon emissions by figuring out the best way to use their 30 essential items. The problem is divided into two parts. The first part is about minimizing the total carbon emissions given that each item must be used at least 5 times a year. The second part introduces the possibility of sharing 10 of these items with roommates, which reduces the effective usage for each shared item.Starting with the first part. The carbon emission for each item is modeled by the function ( C(x) = frac{1}{1 + e^{-0.1x}} ). So, for each item, the more it's used, the higher the carbon emission, but it's a sigmoid function, meaning it has an S-shape. That means initially, as usage increases, the emissions increase rapidly, but after a certain point, the increase in emissions slows down.Alex wants to minimize the total carbon emissions, so we need to minimize the sum of ( C(x_i) ) for all 30 items. The constraint is that each ( x_i ) must be at least 5. So, we have to find ( x_1, x_2, ldots, x_{30} ) such that each ( x_i geq 5 ) and the sum ( sum_{i=1}^{30} frac{1}{1 + e^{-0.1x_i}} ) is minimized.Hmm, okay. So, this is an optimization problem with 30 variables and 30 inequality constraints. Since all the constraints are the same (each ( x_i geq 5 )), and the objective function is the sum of individual functions, maybe we can analyze one item first and then generalize.Let's consider a single item. The function ( C(x) = frac{1}{1 + e^{-0.1x}} ) is increasing because as ( x ) increases, ( e^{-0.1x} ) decreases, so the denominator decreases, making the whole fraction increase. Therefore, to minimize ( C(x) ), we need to minimize ( x ). But since ( x ) has a lower bound of 5, the minimal ( C(x) ) for each item is achieved when ( x_i = 5 ).Wait, that seems straightforward. If each function is increasing, then the minimal total would be achieved by setting each ( x_i ) to its minimum value, which is 5. So, for all 30 items, set ( x_i = 5 ). That would give the minimal total carbon emissions.But let me double-check. Suppose we have two items. If we set both to 5, the total emissions are ( 2 times frac{1}{1 + e^{-0.5}} ). If we set one to 5 and the other to, say, 6, the total emissions would be ( frac{1}{1 + e^{-0.5}} + frac{1}{1 + e^{-0.6}} ). Since ( frac{1}{1 + e^{-0.6}} > frac{1}{1 + e^{-0.5}} ), the total would be higher. So, indeed, setting each ( x_i ) to the minimum usage reduces the total emissions.Therefore, for the first part, the optimal solution is to set each ( x_i = 5 ).Moving on to the second part. Alex can share 10 of these items with roommates, which reduces the effective usage to ( 0.6x_i ) for each shared item. So, for these 10 items, the carbon emission becomes ( C(0.6x_i) = frac{1}{1 + e^{-0.1 times 0.6x_i}} = frac{1}{1 + e^{-0.06x_i}} ).Wait, hold on. Is the usage ( x_i ) the same as before, but the effective usage is 0.6 times that? Or is the effective usage 0.6 times the original? The problem says, "the effective usage by Alex decreases to ( 0.6x_i ) for each shared item ( i )." So, I think it means that instead of using the item ( x_i ) times, the effective usage is 0.6 times ( x_i ). So, the carbon emission for each shared item is ( C(0.6x_i) ).But we still have the constraint that each item must be used a minimum of 5 times a year. So, for the shared items, the effective usage is 0.6x_i, but the actual usage ( x_i ) must still be at least 5. So, the constraint remains ( x_i geq 5 ) for all items, including the shared ones.But wait, if the effective usage is 0.6x_i, does that mean that the carbon emission is calculated based on 0.6x_i? So, for shared items, the carbon emission is ( C(0.6x_i) ), and for non-shared items, it's ( C(x_i) ).So, now, we have 10 shared items and 20 non-shared items. For each shared item, the carbon emission is ( frac{1}{1 + e^{-0.06x_i}} ), and for non-shared, it's ( frac{1}{1 + e^{-0.1x_i}} ).Our goal is still to minimize the total carbon emissions, which is the sum over all 30 items, with the first 10 being shared and the next 20 being non-shared. Each ( x_i geq 5 ).So, similar to before, we need to find ( x_1, x_2, ldots, x_{30} ) such that each ( x_i geq 5 ), and the sum ( sum_{i=1}^{10} frac{1}{1 + e^{-0.06x_i}} + sum_{i=11}^{30} frac{1}{1 + e^{-0.1x_i}} ) is minimized.Again, for each item, whether shared or not, the carbon emission function is increasing in ( x_i ). For shared items, the function is ( C(0.6x_i) = frac{1}{1 + e^{-0.06x_i}} ), which is also increasing because as ( x_i ) increases, ( e^{-0.06x_i} ) decreases, so the denominator decreases, making the whole fraction increase.Therefore, similar to the first part, to minimize the total carbon emissions, we should set each ( x_i ) to its minimum value, which is 5. So, for all 30 items, set ( x_i = 5 ).Wait, but for the shared items, even though the effective usage is 0.6x_i, the constraint is still on the actual usage ( x_i ). So, setting ( x_i = 5 ) for shared items would result in effective usage of 3, but the constraint is on the actual usage, not the effective one. So, as long as ( x_i geq 5 ), it's fine, regardless of the effective usage.Therefore, the optimal solution remains the same: set all ( x_i = 5 ). But wait, does sharing affect the minimal total emissions? Let me think.If we set ( x_i = 5 ) for all items, the total emissions would be 10 times ( frac{1}{1 + e^{-0.3}} ) plus 20 times ( frac{1}{1 + e^{-0.5}} ). If we instead set some ( x_i ) higher for shared items, would that reduce the total emissions? Wait, no, because for each shared item, increasing ( x_i ) would increase ( C(0.6x_i) ), which is bad. So, to minimize, we still set ( x_i = 5 ).But wait, is there a way to have some items used more and others less? For example, if we can reallocate usage between shared and non-shared items to reduce the total emissions. But since all items have the same minimal usage constraint, and each function is increasing, there's no benefit in increasing any ( x_i ) beyond the minimum.Therefore, the optimal solution is still to set all ( x_i = 5 ).Wait, but maybe I'm missing something. If some items are shared, their effective usage is lower, so maybe we can use them more without increasing emissions as much? But no, because the constraint is on the actual usage, not the effective one. So, even if the effective usage is lower, the actual usage still has to be at least 5. So, increasing actual usage beyond 5 would only increase the carbon emission for that item, which is counterproductive.Therefore, the optimal usage pattern remains ( x_i = 5 ) for all items, whether shared or not.But let me verify this by considering the derivative of the carbon emission function. For a single item, the derivative of ( C(x) ) with respect to ( x ) is ( C'(x) = frac{0.1e^{-0.1x}}{(1 + e^{-0.1x})^2} ). This derivative is always positive, meaning the function is increasing. Similarly, for shared items, the derivative is ( C'(0.6x) times 0.6 = frac{0.06e^{-0.06x}}{(1 + e^{-0.06x})^2} ), which is also positive. Therefore, increasing ( x ) beyond 5 will always increase the carbon emission for each item, so the minimal total is achieved at ( x_i = 5 ).So, in conclusion, both for the first and second parts, the optimal usage pattern is to set each item's usage to the minimum required, which is 5 times a year. Sharing the items doesn't change this conclusion because the constraint is on the actual usage, not the effective one, and increasing usage beyond the minimum only increases emissions.Wait, but hold on. If sharing reduces the effective usage, maybe Alex can use the shared items less, but the constraint is on the actual usage. So, if Alex uses a shared item more, the effective usage is higher, but the constraint is still on the actual usage. So, if Alex uses a shared item more, the effective usage is 0.6 times that, but the actual usage is still above 5. So, perhaps, by using shared items more, Alex can reduce the usage of non-shared items? But no, because the constraint is that each item must be used at least 5 times. So, if Alex uses a shared item more, they can't reduce the usage of non-shared items below 5.Wait, maybe I'm overcomplicating. Since all items must be used at least 5 times, regardless of sharing, the minimal total emissions are achieved when all are used exactly 5 times. Sharing just affects the effective usage, but since the constraint is on the actual usage, it doesn't allow us to reduce the actual usage below 5. Therefore, the optimal solution remains the same.So, to summarize, for both parts, the optimal usage is ( x_i = 5 ) for all items. Sharing doesn't change the minimal total emissions because the constraint is on the actual usage, not the effective one.</think>

🧠:<think>Alright, so I have two parts to solve here: part A and part B. Let me start with part A.Problem A: Budget AllocationProfessor Adisa has a budget of 150,000 to distribute among five African countries: Nigeria, South Africa, Kenya, Ghana, and Ethiopia. The allocation needs to be proportional to the product of the number of mentees and the Legal Complexity Score (LCS) for each country.First, let me list out the given data:- Number of Mentees: - Nigeria: 15 - South Africa: 20 - Kenya: 10 - Ghana: 8 - Ethiopia: 12- Legal Complexity Scores (LCS): - Nigeria: 2.5 - South Africa: 3.0 - Kenya: 1.8 - Ghana: 2.0 - Ethiopia: 2.7So, the allocation for each country is proportional to the product of mentees and LCS. That means for each country, I need to calculate mentees * LCS, then find the total of all these products, and then each country's allocation will be (mentees * LCS) / total * 150,000.Let me compute the product for each country first.1. Nigeria: Mentees = 15 LCS = 2.5 Product = 15 * 2.5 = 37.52. South Africa: Mentees = 20 LCS = 3.0 Product = 20 * 3.0 = 603. Kenya: Mentees = 10 LCS = 1.8 Product = 10 * 1.8 = 184. Ghana: Mentees = 8 LCS = 2.0 Product = 8 * 2.0 = 165. Ethiopia: Mentees = 12 LCS = 2.7 Product = 12 * 2.7 = 32.4Now, let's sum all these products to get the total.Total = 37.5 + 60 + 18 + 16 + 32.4Let me compute that step by step:37.5 + 60 = 97.597.5 + 18 = 115.5115.5 + 16 = 131.5131.5 + 32.4 = 163.9So, the total product is 163.9.Now, each country's allocation is (their product / 163.9) * 150,000.Let me compute each allocation:1. Nigeria: Allocation = (37.5 / 163.9) * 150,000 Let me compute 37.5 / 163.9 first. 37.5 ÷ 163.9 ≈ 0.2289 Then, 0.2289 * 150,000 ≈ 34,3352. South Africa: Allocation = (60 / 163.9) * 150,000 60 ÷ 163.9 ≈ 0.3662 0.3662 * 150,000 ≈ 54,9303. Kenya: Allocation = (18 / 163.9) * 150,000 18 ÷ 163.9 ≈ 0.1099 0.1099 * 150,000 ≈ 16,4854. Ghana: Allocation = (16 / 163.9) * 150,000 16 ÷ 163.9 ≈ 0.0976 0.0976 * 150,000 ≈ 14,6405. Ethiopia: Allocation = (32.4 / 163.9) * 150,000 32.4 ÷ 163.9 ≈ 0.1977 0.1977 * 150,000 ≈ 29,655Let me verify that the total allocations add up to approximately 150,000.Adding up all the allocations:34,335 + 54,930 = 89,26589,265 + 16,485 = 105,750105,750 + 14,640 = 120,390120,390 + 29,655 = 150,045Hmm, that's a bit over 150,000 due to rounding errors. To be precise, maybe I should carry more decimal places in the calculations.Alternatively, perhaps I can compute each allocation more accurately.Let me try recalculating with more precise decimal places.First, compute each country's ratio:1. Nigeria: 37.5 / 163.9 ≈ 0.228875 0.228875 * 150,000 = 34,331.252. South Africa: 60 / 163.9 ≈ 0.366183 0.366183 * 150,000 ≈ 54,927.453. Kenya: 18 / 163.9 ≈ 0.109884 0.109884 * 150,000 ≈ 16,482.604. Ghana: 16 / 163.9 ≈ 0.097585 0.097585 * 150,000 ≈ 14,637.755. Ethiopia: 32.4 / 163.9 ≈ 0.197713 0.197713 * 150,000 ≈ 29,656.95Now, let's add these more precise numbers:34,331.25 + 54,927.45 = 89,258.7089,258.70 + 16,482.60 = 105,741.30105,741.30 + 14,637.75 = 120,379.05120,379.05 + 29,656.95 = 150,036.00Still, it's slightly over 150,000. This is due to rounding each allocation. To make it precise, perhaps we can adjust one of the allocations to make the total exactly 150,000.Alternatively, since the total is 163.9, and 150,000 / 163.9 ≈ 915.31 per unit.Wait, maybe another approach: instead of multiplying each ratio by 150,000, perhaps compute the allocation as (mentees * LCS) * (150,000 / total).Which is the same as above, but perhaps if I compute 150,000 / 163.9 first, which is approximately 915.31 per unit.So, each unit of (mentees * LCS) is worth approximately 915.31.Therefore:1. Nigeria: 37.5 * 915.31 ≈ 34,331.252. South Africa: 60 * 915.31 ≈ 54,918.603. Kenya: 18 * 915.31 ≈ 16,475.584. Ghana: 16 * 915.31 ≈ 14,644.965. Ethiopia: 32.4 * 915.31 ≈ 29,656.95Adding these:34,331.25 + 54,918.60 = 89,249.8589,249.85 + 16,475.58 = 105,725.43105,725.43 + 14,644.96 = 120,370.39120,370.39 + 29,656.95 = 150,027.34Still, it's a bit over. Maybe due to the approximation of 915.31. Let me compute 150,000 / 163.9 more accurately.150,000 ÷ 163.9 ≈ 915.3148So, using this more precise multiplier:1. Nigeria: 37.5 * 915.3148 ≈ 34,331.052. South Africa: 60 * 915.3148 ≈ 54,918.893. Kenya: 18 * 915.3148 ≈ 16,475.674. Ghana: 16 * 915.3148 ≈ 14,645.045. Ethiopia: 32.4 * 915.3148 ≈ 29,656.95Adding these:34,331.05 + 54,918.89 = 89,249.9489,249.94 + 16,475.67 = 105,725.61105,725.61 + 14,645.04 = 120,370.65120,370.65 + 29,656.95 = 150,027.60Still, it's about 150,027.60, which is 27.60 over. To fix this, perhaps we can subtract a little from one of the allocations.Alternatively, since the difference is minimal, maybe it's acceptable to have a slight rounding difference.But perhaps a better approach is to calculate each allocation with more precise decimal places without rounding until the end.Alternatively, use fractions.But that might be too time-consuming.Alternatively, accept that due to rounding, the total might be slightly over or under, but in reality, the exact allocation would sum to 150,000.But for the purposes of this problem, I think the initial approximate allocations are acceptable, acknowledging that rounding might cause a slight discrepancy.So, summarizing:- Nigeria: ~34,331- South Africa: ~54,919- Kenya: ~16,476- Ghana: ~14,645- Ethiopia: ~29,657Let me check if these add up:34,331 + 54,919 = 89,25089,250 + 16,476 = 105,726105,726 + 14,645 = 120,371120,371 + 29,657 = 150,028Still, it's 28 over. Hmm.Alternatively, maybe I should carry more decimal places in the initial ratio.Let me compute each ratio with more precision.First, total = 163.9So, 150,000 / 163.9 = ?Let me compute 150,000 ÷ 163.9.163.9 goes into 150,000 how many times?163.9 * 915 = 163.9 * 900 = 147,510; 163.9 * 15 = 2,458.5; total 147,510 + 2,458.5 = 149,968.5So, 163.9 * 915 = 149,968.5Subtract from 150,000: 150,000 - 149,968.5 = 31.5So, 31.5 / 163.9 ≈ 0.192So, 915 + 0.192 ≈ 915.192So, the multiplier is approximately 915.192 per unit.Therefore, recalculating each allocation:1. Nigeria: 37.5 * 915.192 ≈ 37.5 * 915 + 37.5 * 0.192 37.5 * 915 = 34,312.5 37.5 * 0.192 = 7.2 Total ≈ 34,312.5 + 7.2 = 34,319.7 ≈ 34,3202. South Africa: 60 * 915.192 ≈ 60 * 915 + 60 * 0.192 60 * 915 = 54,900 60 * 0.192 = 11.52 Total ≈ 54,900 + 11.52 = 54,911.52 ≈ 54,911.523. Kenya: 18 * 915.192 ≈ 18 * 915 + 18 * 0.192 18 * 915 = 16,470 18 * 0.192 = 3.456 Total ≈ 16,470 + 3.456 = 16,473.456 ≈ 16,473.464. Ghana: 16 * 915.192 ≈ 16 * 915 + 16 * 0.192 16 * 915 = 14,640 16 * 0.192 = 3.072 Total ≈ 14,640 + 3.072 = 14,643.072 ≈ 14,643.075. Ethiopia: 32.4 * 915.192 ≈ Let's compute 32 * 915.192 + 0.4 * 915.192 32 * 915.192 = 29,286.144 0.4 * 915.192 = 366.0768 Total ≈ 29,286.144 + 366.0768 = 29,652.2208 ≈ 29,652.22Now, let's add these precise allocations:34,320 + 54,911.52 = 89,231.5289,231.52 + 16,473.46 = 105,704.98105,704.98 + 14,643.07 = 120,348.05120,348.05 + 29,652.22 = 150,000.27Wow, that's very close to 150,000.27, which is just 27 cents over. That's acceptable due to rounding.So, the allocations are approximately:- Nigeria: 34,320- South Africa: 54,911.52- Kenya: 16,473.46- Ghana: 14,643.07- Ethiopia: 29,652.22But since we're dealing with currency, it's better to round to the nearest dollar or perhaps to two decimal places.So, rounding each to the nearest dollar:- Nigeria: 34,320- South Africa: 54,912- Kenya: 16,473- Ghana: 14,643- Ethiopia: 29,652Let me check the total:34,320 + 54,912 = 89,23289,232 + 16,473 = 105,705105,705 + 14,643 = 120,348120,348 + 29,652 = 150,000Perfect! Now, the total is exactly 150,000.So, the allocations are:- Nigeria: 34,320- South Africa: 54,912- Kenya: 16,473- Ghana: 14,643- Ethiopia: 29,652Alternatively, if we need to present them with cents, we can keep the decimal values, but since the total is exact when rounded to the dollar, it's fine.Problem B: Finding the Average kProfessor Adisa models the improvement in human rights outcomes with the function ( f(x) = 100 left(1 - e^{-kx}right) ), where ( x ) is the number of years since the reform, and ( k ) is a country-specific constant. The average observed improvement in the first year (x=1) for all countries is 25%. We need to find the average value of ( k ) across all countries.First, let's understand the function. At x=1, the improvement is 25%, so:( f(1) = 100(1 - e^{-k}) = 25 )So, 100(1 - e^{-k}) = 25Divide both sides by 100:1 - e^{-k} = 0.25Therefore:e^{-k} = 1 - 0.25 = 0.75Take the natural logarithm of both sides:ln(e^{-k}) = ln(0.75)Simplify:-k = ln(0.75)Multiply both sides by -1:k = -ln(0.75)Compute ln(0.75):ln(0.75) ≈ -0.28768207So, k ≈ 0.28768207But wait, this is for each country. However, the problem states that the average observed improvement in the first year for all countries is 25%. So, does this mean that each country has the same k, or do they have different k's, and the average of f(1) across countries is 25%?Wait, the function is given as ( f(x) = 100(1 - e^{-kx}) ). For each country, k is a constant. The average improvement in the first year across all countries is 25%. So, if there are five countries, each with their own k_i, then the average of f(1) for each country is 25%.So, mathematically:( f_1(1) + f_2(1) + f_3(1) + f_4(1) + f_5(1) ) / 5 = 25Where f_i(1) = 100(1 - e^{-k_i * 1}) for each country i.So, sum_{i=1 to 5} [100(1 - e^{-k_i})] / 5 = 25Multiply both sides by 5:sum_{i=1 to 5} [100(1 - e^{-k_i})] = 125Divide both sides by 100:sum_{i=1 to 5} (1 - e^{-k_i}) = 1.25So, sum_{i=1 to 5} (1 - e^{-k_i}) = 1.25Which implies:sum_{i=1 to 5} 1 - sum_{i=1 to 5} e^{-k_i} = 1.25sum_{i=1 to 5} 1 is 5, so:5 - sum_{i=1 to 5} e^{-k_i} = 1.25Therefore:sum_{i=1 to 5} e^{-k_i} = 5 - 1.25 = 3.75So, sum_{i=1 to 5} e^{-k_i} = 3.75We need to find the average value of k across all countries, i.e., (k1 + k2 + k3 + k4 + k5)/5.But we have sum_{i=1 to 5} e^{-k_i} = 3.75This seems tricky because we have an equation involving the sum of exponentials of k_i, and we need the average of k_i. Without more information about the individual k_i's, we can't directly compute the average k.However, perhaps we can assume that all k_i are equal. If that's the case, then each e^{-k} would be equal, and we can solve for k.Let me check if that's a valid assumption.If all k_i = k, then:5 * e^{-k} = 3.75So, e^{-k} = 3.75 / 5 = 0.75Which is the same as before.So, k = -ln(0.75) ≈ 0.28768207Therefore, if all k_i are equal, then each k is approximately 0.2877, and the average k is also 0.2877.But the problem doesn't specify whether the k's are the same or different. It just says "the average observed improvement in the first year for all countries is 25%". So, it's possible that each country has a different k, but the average f(1) is 25%.However, without additional information about the distribution of k's or the individual f(1)'s, we can't determine the exact average k. But if we assume that all k's are equal, which is a common assumption in such problems unless stated otherwise, then we can find k ≈ 0.2877.Alternatively, if the k's are different, we can't find the exact average k without more data. But since the problem asks for the average value of k across all countries, and given that the average f(1) is 25%, it's likely that we're supposed to assume that each country has the same k, leading to k ≈ 0.2877.Therefore, the average k is approximately 0.2877.But let me verify this assumption.If all k's are equal, then yes, each k is 0.2877, and the average is the same.If the k's are different, then the average k could be different. For example, suppose one country has a very high k, making f(1) close to 100%, and another has a very low k, making f(1) close to 0%, but their average is still 25%. However, without knowing the distribution, we can't compute the average k.But in the absence of specific information, the most straightforward assumption is that all k's are equal, leading to k ≈ 0.2877.Therefore, the average k is approximately 0.2877.To express this more precisely, since ln(0.75) = -0.28768207, so k = 0.28768207.Rounding to four decimal places, k ≈ 0.2877.Alternatively, if we want to express it as a fraction, since 0.2877 is approximately 0.2877 ≈ 0.2877 ≈ 0.2877, but it's a transcendental number, so it's better to leave it as is.So, the average k is approximately 0.2877.But let me check the math again.Given that the average f(1) is 25%, so:( f1(1) + f2(1) + f3(1) + f4(1) + f5(1) ) / 5 = 25Which implies:sum_{i=1 to 5} f_i(1) = 125Each f_i(1) = 100(1 - e^{-k_i})So, sum_{i=1 to 5} [100(1 - e^{-k_i})] = 125Divide both sides by 100:sum_{i=1 to 5} (1 - e^{-k_i}) = 1.25Which simplifies to:5 - sum_{i=1 to 5} e^{-k_i} = 1.25Thus:sum_{i=1 to 5} e^{-k_i} = 3.75If we assume all k_i are equal, then:5 * e^{-k} = 3.75 => e^{-k} = 0.75 => k = -ln(0.75) ≈ 0.28768207So, yes, that's correct.Therefore, the average k is approximately 0.2877.But to express it more precisely, we can write it as:k = ln(4/3) ≈ 0.28768207Because 0.75 = 3/4, so ln(4/3) is the exact value.Therefore, the average k is ln(4/3).So, ln(4/3) is approximately 0.28768207.Hence, the average k is ln(4/3).Final AnswerA) The allocations are:- Nigeria: boxed{34320}- South Africa: boxed{54912}- Kenya: boxed{16473}- Ghana: boxed{14643}- Ethiopia: boxed{29652}B) The average value of ( k ) is boxed{lnleft(frac{4}{3}right)}.</think>