🧠:<think>Okay, so I need to calculate the intrinsic value of XYZ Corp's stock using the DCF model. Hmm, let me recall what DCF is. It's a valuation method that estimates the value of an investment based on its expected future cash flows, discounted at a rate that reflects the risk of those cash flows. The problem gives me the yearly free cash flows for the last five years: 120, 140, 170, 200, 230 million. It also mentions that the free cash flow is expected to grow at a constant rate of 5% per year indefinitely starting from the last observed year. The discount rate is 8%. Alright, so first, I need to figure out the terminal value because the cash flows are growing perpetually at 5%. The formula for the terminal value in a DCF model is usually the cash flow in the last year multiplied by (1 + growth rate) divided by (discount rate - growth rate). Wait, let me write that down to make sure I get it right. The terminal value (TV) is calculated as:TV = FCF_n * (1 + g) / (r - g)Where:- FCF_n is the free cash flow in the last year of the forecast period- g is the perpetual growth rate- r is the discount rateIn this case, the last observed free cash flow is 230 million, and the growth rate is 5%, so g = 0.05. The discount rate r is 8%, so r = 0.08.So plugging in the numbers:TV = 230 * (1 + 0.05) / (0.08 - 0.05)TV = 230 * 1.05 / 0.03TV = 230 * 35Wait, 1.05 divided by 0.03 is 35? Let me check that. 0.03 goes into 1.05 exactly 35 times because 0.03 * 35 = 1.05. So yes, that's correct.So TV = 230 * 35 = 8,050 million.Now, I need to discount all the free cash flows back to the present value. The formula for the present value of each cash flow is FCF / (1 + r)^t, where t is the year.But wait, the terminal value is for year 5, right? So I need to discount each of the five cash flows and then the terminal value.Let me list out the cash flows:Year 1: 120 millionYear 2: 140 millionYear 3: 170 millionYear 4: 200 millionYear 5: 230 millionTerminal Value: 8,050 million at the end of Year 5.So I need to calculate the present value (PV) for each of these.Let me compute each PV step by step.First, PV of Year 1 cash flow:PV1 = 120 / (1 + 0.08)^1 = 120 / 1.08 ≈ 111.11 millionPV2 = 140 / (1.08)^2 = 140 / 1.1664 ≈ 120.00 millionWait, 1.08 squared is 1.1664, right? Yes. So 140 divided by 1.1664 is approximately 120.00.PV3 = 170 / (1.08)^3. Let's compute 1.08 cubed. 1.08 * 1.08 = 1.1664, then 1.1664 * 1.08 ≈ 1.2597. So PV3 = 170 / 1.2597 ≈ 135.00 million.PV4 = 200 / (1.08)^4. Let's compute 1.08^4. 1.08^3 is about 1.2597, so 1.2597 * 1.08 ≈ 1.3605. So PV4 = 200 / 1.3605 ≈ 147.00 million.PV5 = 230 / (1.08)^5. 1.08^5 is approximately 1.4693. So PV5 = 230 / 1.4693 ≈ 156.60 million.Now, the terminal value is at Year 5, so its PV is TV / (1.08)^5.We already calculated TV as 8,050 million. So PV of TV = 8,050 / 1.4693 ≈ 5,480.00 million.Wait, let me verify that division: 8,050 divided by 1.4693. Let me compute 8,050 / 1.4693.1.4693 * 5,000 = 7,346.5Subtract that from 8,050: 8,050 - 7,346.5 = 703.5Now, 1.4693 * 480 ≈ 705.264So 5,000 + 480 = 5,480, which gives us approximately 7,346.5 + 705.264 ≈ 8,051.764, which is very close to 8,050. So yes, approximately 5,480 million.Now, sum up all the present values:PV1 ≈ 111.11PV2 ≈ 120.00PV3 ≈ 135.00PV4 ≈ 147.00PV5 ≈ 156.60PV of TV ≈ 5,480.00Total PV = 111.11 + 120 + 135 + 147 + 156.6 + 5,480Let me add them step by step:111.11 + 120 = 231.11231.11 + 135 = 366.11366.11 + 147 = 513.11513.11 + 156.6 = 669.71669.71 + 5,480 = 6,149.71 millionSo the total present value of all cash flows is approximately 6,149.71 million.But wait, is that the total value of the firm? Or do I need to adjust for anything else? Hmm, in DCF, the total value is the sum of the present values of all free cash flows, including the terminal value. So yes, that's the total value.Now, to get the intrinsic value per share, I need to divide this total value by the number of outstanding shares, which is 10 million.So intrinsic value per share = 6,149.71 / 10 ≈ 614.97 per share.Wait, that seems really high. The current market price is 55 per share. So according to this, the intrinsic value is about 615, which is way higher than the market price. So the stock is underpriced.But wait, let me double-check my calculations because 615 seems extremely high. Maybe I made a mistake in calculating the terminal value or the discounting.Let me go back.Terminal value was calculated as 230 * 1.05 / (0.08 - 0.05) = 230 * 1.05 / 0.03 = 230 * 35 = 8,050 million. That seems correct.Then, discounting each cash flow:Year 1: 120 / 1.08 = ~111.11Year 2: 140 / 1.1664 ≈ 120Year 3: 170 / 1.2597 ≈ 135Year 4: 200 / 1.3605 ≈ 147Year 5: 230 / 1.4693 ≈ 156.6Terminal value: 8,050 / 1.4693 ≈ 5,480Adding up: 111.11 + 120 + 135 + 147 + 156.6 + 5,480 ≈ 6,149.71 millionDivide by 10 million shares: ~615 per share.Hmm, that seems correct, but maybe the growth rate assumption is too high? Or perhaps the free cash flows are too high. Alternatively, maybe I should use the free cash flow to equity instead of total free cash flow? Wait, the problem says "Free Cash Flow" without specifying, but in DCF, usually, we use free cash flow to the firm (FCFF) or free cash flow to equity (FCFE). Since it's not specified, I assumed FCFF, which is appropriate for the firm valuation.Alternatively, maybe the question expects us to use the last free cash flow as the starting point for the terminal value without growing it once? Wait, no, the terminal value formula already incorporates the growth rate.Wait, another thought: sometimes, the terminal value is calculated as FCF_n * (1 + g) / (r - g), which is what I did. So that's correct.Alternatively, maybe the question expects the terminal value to be calculated at the end of year 5, so we don't need to discount it further? No, because in DCF, all cash flows are discounted back to the present. So the terminal value is at year 5, so we need to discount it back 5 years.Wait, let me check the calculation of the terminal value again. 230 * 1.05 = 241.5. Then 241.5 / (0.08 - 0.05) = 241.5 / 0.03 = 8,050. So that's correct.Hmm, maybe the issue is that the free cash flows are in millions, and the terminal value is also in millions, so when we sum them up, it's in millions. Then dividing by 10 million shares gives us the per share value.Alternatively, maybe I should have used the free cash flow to equity instead. But since the problem doesn't specify, I think FCFF is acceptable.Alternatively, perhaps the question expects the growth rate to be applied to the last cash flow without multiplying by (1 + g). Let me check that.If I don't multiply by (1 + g), then TV = 230 / (0.08 - 0.05) = 230 / 0.03 ≈ 7,666.67 million. Then PV of TV = 7,666.67 / 1.4693 ≈ 5,222.22 million.Then total PV would be:111.11 + 120 + 135 + 147 + 156.6 + 5,222.22 ≈ 5,891.93 millionDivide by 10 million shares: ~589.19 per share.Still, that's way higher than 55. So regardless, the intrinsic value is much higher than the market price.Wait, but maybe I made a mistake in the discounting of the terminal value. Let me think again.Terminal value is calculated at year 5, so to get its present value, we discount it by (1 + r)^5, which is 1.4693. So 8,050 / 1.4693 ≈ 5,480. That seems correct.Alternatively, maybe the question expects the terminal value to be calculated as FCF_n / (r - g) without the (1 + g) factor. Let me try that.So TV = 230 / (0.08 - 0.05) = 230 / 0.03 ≈ 7,666.67 million.Then PV of TV = 7,666.67 / 1.4693 ≈ 5,222.22 million.Total PV:111.11 + 120 + 135 + 147 + 156.6 + 5,222.22 ≈ 5,891.93 million.Still, that's about 589 per share.Wait, but in the standard DCF model, the terminal value is calculated as FCF_n * (1 + g) / (r - g). So I think my initial calculation was correct.Alternatively, maybe the question expects the growth rate to be applied to the last cash flow for the terminal value, but not compounded. Hmm, I'm not sure. Maybe I should check the formula again.Yes, the terminal value formula is FCF_n * (1 + g) / (r - g). So I think my calculation is correct.So, the intrinsic value per share is approximately 615, which is way higher than the current market price of 55. So the stock is underpriced by a significant margin.Wait, but let me think again. Maybe I should have used free cash flow to equity instead of FCFF. Because FCFF is for the entire firm, but if we're valuing equity, we need to subtract debt and add cash. But the problem doesn't provide information on debt or cash, so maybe it's assuming FCFF is the same as FCFE, which is not usually the case. But since the problem doesn't specify, I think we have to proceed with the given data.Alternatively, maybe the question expects us to use the free cash flow as is, without considering the terminal value. But no, the question says to assume constant growth indefinitely, so terminal value is necessary.Alternatively, maybe I should have used the net income instead of free cash flow? But the question specifically says to use free cash flow, so that's correct.Wait, another thought: maybe the discount rate should be adjusted for the risk of the cash flows. But the problem gives a discount rate of 8%, so we don't need to adjust it.Alternatively, perhaps I made a mistake in the calculation of the present values. Let me recalculate each PV.Year 1: 120 / 1.08 = 111.111 millionYear 2: 140 / (1.08)^2 = 140 / 1.1664 ≈ 120.00 millionYear 3: 170 / (1.08)^3 = 170 / 1.259712 ≈ 135.00 millionYear 4: 200 / (1.08)^4 = 200 / 1.36048896 ≈ 147.00 millionYear 5: 230 / (1.08)^5 = 230 / 1.469328077 ≈ 156.60 millionTerminal value: 8,050 / 1.469328077 ≈ 5,480.00 millionAdding them up:111.111 + 120 = 231.111231.111 + 135 = 366.111366.111 + 147 = 513.111513.111 + 156.6 = 669.711669.711 + 5,480 = 6,149.711 millionYes, that's correct.So, intrinsic value per share is 6,149.711 / 10 ≈ 614.97, which is approximately 615.So, the intrinsic value is 615, and the market price is 55. So the stock is underpriced.To find the percentage difference, we can calculate (Intrinsic Value - Market Price) / Market Price * 100%.So, (615 - 55) / 55 * 100% ≈ (560) / 55 * 100% ≈ 10.18 * 100% ≈ 1,018.18% underpriced.Wait, that seems extremely high. Is that correct? Let me check the calculation.(615 - 55) = 560560 / 55 ≈ 10.181810.1818 * 100 ≈ 1,018.18%Yes, that's correct. So the stock is underpriced by approximately 1,018%.But that seems unrealistic. Maybe I made a mistake in the terminal value calculation. Let me think again.Wait, the free cash flows are in millions, and the terminal value is also in millions. So when I sum up all the PVs, it's in millions, and then dividing by 10 million shares gives me the per share value.Alternatively, maybe the question expects the terminal value to be calculated differently, or perhaps the growth rate is applied differently.Wait, another approach: sometimes, the terminal value is calculated as FCF_n / (r - g), without multiplying by (1 + g). Let me try that.So, TV = 230 / (0.08 - 0.05) = 230 / 0.03 ≈ 7,666.67 million.Then PV of TV = 7,666.67 / 1.4693 ≈ 5,222.22 million.Total PV:111.11 + 120 + 135 + 147 + 156.6 + 5,222.22 ≈ 5,891.93 million.Divide by 10 million shares: ~589.19 per share.Still, that's way higher than 55. So the percentage difference would be (589.19 - 55) / 55 * 100% ≈ (534.19) / 55 ≈ 9.7125 * 100 ≈ 971.25%.Still, that's a huge percentage. Maybe the issue is that the free cash flows are growing at 5% indefinitely, but the discount rate is only 8%, which is not that high, so the terminal value becomes very large.Alternatively, maybe the question expects us to use a different method, like the Gordon Growth Model, but applied to free cash flow. But that's essentially what I did.Wait, another thought: maybe the question expects us to use the free cash flow to equity, which would require adjusting for debt and cash. But since we don't have that information, I think we have to proceed with the given data.Alternatively, perhaps the question expects us to use the free cash flow as is, without considering the terminal value beyond year 5. But that's not the case because it says to assume constant growth indefinitely.Alternatively, maybe I should have used the free cash flow in year 5 as the base for the terminal value without growing it once. Wait, that's what I did in the second calculation, but it still resulted in a high intrinsic value.Wait, perhaps the question expects us to use the free cash flow in year 5 as the base without multiplying by (1 + g). So TV = FCF5 / (r - g) = 230 / (0.08 - 0.05) = 7,666.67 million. Then PV of TV = 7,666.67 / 1.4693 ≈ 5,222.22 million.Total PV: 111.11 + 120 + 135 + 147 + 156.6 + 5,222.22 ≈ 5,891.93 million.Divide by 10 million shares: ~589.19 per share.Still, that's way higher than 55.Alternatively, maybe the question expects us to use the free cash flow in year 5 as the base and not discount the terminal value? That would be incorrect because terminal value is a future cash flow and needs to be discounted.Alternatively, maybe the question expects us to use a different discount rate for the terminal value? But the problem states a discount rate of 8%, so that's the rate to use.Alternatively, perhaps I should have used the free cash flow in year 5 as the base and then applied the growth rate for the terminal value, but not discount it. That would be incorrect because all future cash flows must be discounted.Wait, another thought: maybe the question expects us to calculate the terminal value as FCF5 * (1 + g) / (r - g), which is what I did, but then not discount it because it's already at year 5. But no, the terminal value is at year 5, so we need to discount it back to the present.Wait, let me think about the timeline. The free cash flows are for years 1 to 5, and the terminal value is at year 5. So the terminal value is the value of all cash flows from year 6 onwards, discounted back to year 5. Then, to get the present value, we discount it back to year 0.So, yes, the terminal value is at year 5, so we need to discount it by (1 + r)^5.Therefore, my initial calculation is correct.So, the intrinsic value per share is approximately 615, which is way higher than the market price of 55. Therefore, the stock is underpriced by approximately 1,018%.But that seems extremely high. Maybe the question expects a different approach, like using the free cash flow to equity or adjusting for other factors. But since the problem doesn't provide additional information, I think my approach is correct.Alternatively, maybe the question expects us to use the free cash flow in year 5 as the base for the terminal value without growing it once, but that still results in a high intrinsic value.Wait, let me try another approach. Maybe the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 6. So the terminal value would be FCF5 * (1 + g) / (r - g), which is what I did.Alternatively, maybe the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 5, meaning that the terminal value is FCF5 / (r - g). But that would be 230 / (0.08 - 0.05) = 7,666.67 million, which is what I did in the second calculation.But regardless, the intrinsic value is still way higher than the market price.Alternatively, maybe the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 6, but then discount the terminal value correctly.Wait, another thought: perhaps the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 6, so the terminal value is FCF5 * (1 + g) / (r - g), which is 230 * 1.05 / (0.08 - 0.05) = 8,050 million, which is what I did.So, I think my calculation is correct.Therefore, the intrinsic value is approximately 615 per share, and the market price is 55, so the stock is underpriced by approximately 1,018%.But that seems extremely high. Maybe I should present the answer as such, but it's worth noting that in reality, such a high intrinsic value compared to the market price is unusual and might indicate that the assumptions (like the growth rate or discount rate) are not realistic.Alternatively, perhaps the question expects us to use the free cash flow to equity, but without information on debt and cash, we can't adjust for that.Alternatively, maybe the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 6, but then discount the terminal value correctly.Wait, another thought: maybe the question expects us to use the free cash flow in year 5 as the base and then grow it at 5% starting from year 6, so the terminal value is FCF5 * (1 + g) / (r - g), which is 230 * 1.05 / (0.08 - 0.05) = 8,050 million, which is what I did.So, I think my calculation is correct.Therefore, the intrinsic value is approximately 615 per share, and the market price is 55, so the stock is underpriced by approximately 1,018%.But let me check the calculation of the percentage difference again.(615 - 55) / 55 = 560 / 55 ≈ 10.1818, which is approximately 1,018.18%.Yes, that's correct.So, in conclusion, the intrinsic value is approximately 615 per share, which is much higher than the market price of 55, indicating that the stock is underpriced by about 1,018%.</think>

🧠:<think>Okay, so I have this problem about the logistic growth model for an octopus population. Let me try to work through it step by step. First, part 1 is asking me to solve the logistic differential equation given the initial population P(0) = P0. I remember that the logistic equation is a common model for population growth where growth slows as the population approaches the carrying capacity K. The equation is:dP/dt = rP(1 - P/K)I think the standard solution to this is a function that starts with exponential growth and then levels off as it approaches K. I recall that the solution involves integrating both sides, maybe using separation of variables. Let me try that.So, starting with:dP/dt = rP(1 - P/K)I can rewrite this as:dP / [P(1 - P/K)] = r dtHmm, to integrate the left side, I might need partial fractions. Let me set up partial fractions for 1/[P(1 - P/K)]. Let me denote 1/[P(1 - P/K)] as A/P + B/(1 - P/K). Multiplying both sides by P(1 - P/K):1 = A(1 - P/K) + BPLet me solve for A and B. Let's plug in P = 0:1 = A(1 - 0) + B(0) => A = 1Now plug in P = K:1 = A(1 - K/K) + B(K) => 1 = A(0) + BK => BK = 1 => B = 1/KSo, the partial fractions decomposition is:1/P + (1/K)/(1 - P/K)Therefore, the integral becomes:∫ [1/P + (1/K)/(1 - P/K)] dP = ∫ r dtLet me compute the integrals.First integral: ∫ 1/P dP = ln|P| + CSecond integral: ∫ (1/K)/(1 - P/K) dP. Let me make a substitution. Let u = 1 - P/K, then du = -1/K dP, so -du = (1/K) dP. Therefore, the integral becomes ∫ (1/K)/(u) * (-K du) = -∫ 1/u du = -ln|u| + C = -ln|1 - P/K| + CSo putting it all together:ln|P| - ln|1 - P/K| = rt + CCombine the logs:ln(P / (1 - P/K)) = rt + CExponentiate both sides:P / (1 - P/K) = e^{rt + C} = e^C e^{rt}Let me denote e^C as another constant, say C1.So:P / (1 - P/K) = C1 e^{rt}Now, solve for P.Multiply both sides by (1 - P/K):P = C1 e^{rt} (1 - P/K)Expand the right side:P = C1 e^{rt} - (C1 e^{rt} P)/KBring the term with P to the left:P + (C1 e^{rt} P)/K = C1 e^{rt}Factor out P:P [1 + (C1 e^{rt})/K] = C1 e^{rt}Therefore:P = [C1 e^{rt}] / [1 + (C1 e^{rt})/K]Simplify denominator:Multiply numerator and denominator by K:P = [C1 K e^{rt}] / [K + C1 e^{rt}]Now, apply the initial condition P(0) = P0.At t=0, P = P0:P0 = [C1 K e^{0}] / [K + C1 e^{0}] = [C1 K] / [K + C1]Multiply both sides by denominator:P0 (K + C1) = C1 KExpand:P0 K + P0 C1 = C1 KBring terms with C1 to one side:P0 C1 - C1 K = -P0 KFactor C1:C1 (P0 - K) = -P0 KTherefore:C1 = (-P0 K)/(P0 - K) = (P0 K)/(K - P0)So, substitute back into P(t):P(t) = [ (P0 K)/(K - P0) * K e^{rt} ] / [ K + (P0 K)/(K - P0) e^{rt} ]Simplify numerator and denominator:Numerator: (P0 K^2)/(K - P0) e^{rt}Denominator: K + (P0 K)/(K - P0) e^{rt} = K [1 + (P0)/(K - P0) e^{rt} ]So, P(t) = [ (P0 K^2)/(K - P0) e^{rt} ] / [ K (1 + (P0)/(K - P0) e^{rt} ) ]Cancel K in numerator and denominator:P(t) = [ (P0 K)/(K - P0) e^{rt} ] / [ 1 + (P0)/(K - P0) e^{rt} ]Let me factor out (P0)/(K - P0) e^{rt} in the denominator:Denominator: 1 + (P0)/(K - P0) e^{rt} = [ (K - P0) + P0 e^{rt} ] / (K - P0)Therefore, P(t) becomes:[ (P0 K)/(K - P0) e^{rt} ] / [ (K - P0 + P0 e^{rt}) / (K - P0) ) ] = [ (P0 K e^{rt} ) / (K - P0) ] * [ (K - P0) / (K - P0 + P0 e^{rt}) ) ]The (K - P0) terms cancel:P(t) = (P0 K e^{rt}) / (K - P0 + P0 e^{rt})I can factor P0 in the denominator:P(t) = (P0 K e^{rt}) / [ K - P0 + P0 e^{rt} ] = (P0 K e^{rt}) / [ K + P0 (e^{rt} - 1) ]Alternatively, sometimes written as:P(t) = K / [1 + (K - P0)/P0 e^{-rt} ]Let me check that. If I take my expression:P(t) = (P0 K e^{rt}) / (K - P0 + P0 e^{rt})Divide numerator and denominator by e^{rt}:P(t) = (P0 K) / ( (K - P0) e^{-rt} + P0 )Which can be written as:P(t) = K / [1 + (K - P0)/P0 e^{-rt} ]Yes, that's another common form. So, both forms are correct. Maybe the second one is more standard.So, that's the solution for part 1.Now, moving on to part 2. The carrying capacity K is now a function of time, K(t) = K0 e^{-α t}. So, the logistic model needs to be modified to incorporate this time-dependent K(t).So, the original logistic equation is:dP/dt = r P (1 - P/K(t))So, substituting K(t) = K0 e^{-α t}, we get:dP/dt = r P (1 - P/(K0 e^{-α t}))So, the differential equation is:dP/dt = r P (1 - P/(K0 e^{-α t}))This is a non-autonomous logistic equation because K is time-dependent.I need to solve this differential equation. Hmm, it's a Bernoulli equation perhaps? Let me see.The standard logistic equation is separable, but with K(t) time-dependent, it's more complicated. Maybe we can use substitution.Let me write the equation as:dP/dt = r P - (r / K0) P^2 e^{α t}So, it's a Riccati equation, which is a type of nonlinear differential equation. Riccati equations are generally difficult to solve unless we have a particular solution.Alternatively, maybe we can use substitution to make it linear.Let me consider substitution: Let Q = 1/P. Then, dQ/dt = -1/P^2 dP/dtSo, plugging into the equation:dQ/dt = -1/P^2 [ r P - (r / K0) P^2 e^{α t} ] = -r / P + (r / K0) e^{α t}But since Q = 1/P, then 1/P = Q, so:dQ/dt = -r Q + (r / K0) e^{α t}So, now we have a linear differential equation in Q:dQ/dt + r Q = (r / K0) e^{α t}Yes, that's linear. So, we can solve this using an integrating factor.The standard form is:dQ/dt + P(t) Q = Q(t)Here, P(t) = r, and Q(t) = (r / K0) e^{α t}The integrating factor μ(t) is e^{∫ r dt} = e^{r t}Multiply both sides by μ(t):e^{r t} dQ/dt + r e^{r t} Q = (r / K0) e^{(α + r) t}The left side is d/dt [ e^{r t} Q ]So, integrate both sides:∫ d/dt [ e^{r t} Q ] dt = ∫ (r / K0) e^{(α + r) t} dtThus,e^{r t} Q = (r / K0) ∫ e^{(α + r) t} dt + CCompute the integral:∫ e^{(α + r) t} dt = e^{(α + r) t} / (α + r) + CSo,e^{r t} Q = (r / K0) * [ e^{(α + r) t} / (α + r) ] + CSimplify:e^{r t} Q = (r / (K0 (α + r))) e^{(α + r) t} + CDivide both sides by e^{r t}:Q = (r / (K0 (α + r))) e^{α t} + C e^{-r t}But Q = 1/P, so:1/P = (r / (K0 (α + r))) e^{α t} + C e^{-r t}Therefore, solving for P:P(t) = 1 / [ (r / (K0 (α + r))) e^{α t} + C e^{-r t} ]Now, apply the initial condition P(0) = P0.At t=0:P0 = 1 / [ (r / (K0 (α + r))) e^{0} + C e^{0} ] = 1 / [ r / (K0 (α + r)) + C ]So,1 / P0 = r / (K0 (α + r)) + CTherefore,C = 1 / P0 - r / (K0 (α + r))So, substitute back into P(t):P(t) = 1 / [ (r / (K0 (α + r))) e^{α t} + (1 / P0 - r / (K0 (α + r))) e^{-r t} ]This expression looks a bit complicated. Maybe we can factor out some terms.Let me write it as:P(t) = 1 / [ A e^{α t} + B e^{-r t} ]Where A = r / (K0 (α + r)) and B = 1 / P0 - r / (K0 (α + r))Alternatively, we can express it in terms of exponentials with different exponents. It might be possible to write it in a more compact form, but perhaps this is sufficient.Alternatively, let me see if I can combine the terms:Let me factor out e^{-r t}:P(t) = 1 / [ A e^{α t} + B e^{-r t} ] = 1 / [ e^{-r t} (A e^{(α + r) t} + B ) ]So,P(t) = e^{r t} / (A e^{(α + r) t} + B )Substitute back A and B:A = r / (K0 (α + r)), B = 1 / P0 - r / (K0 (α + r))So,P(t) = e^{r t} / [ (r / (K0 (α + r))) e^{(α + r) t} + (1 / P0 - r / (K0 (α + r))) ]Simplify the denominator:Denominator = (r / (K0 (α + r))) e^{(α + r) t} + (1 / P0 - r / (K0 (α + r)))Let me factor out r / (K0 (α + r)):= (r / (K0 (α + r))) [ e^{(α + r) t} - 1 ] + 1 / P0Wait, no:Wait, 1 / P0 - r / (K0 (α + r)) is a constant term, so perhaps it's better to leave it as is.Alternatively, perhaps we can write:Denominator = (r / (K0 (α + r))) e^{(α + r) t} + (1 / P0 - r / (K0 (α + r)))= (r e^{(α + r) t} ) / (K0 (α + r)) + (1 / P0 - r / (K0 (α + r)))Let me factor out 1 / (K0 (α + r)):= [ r e^{(α + r) t} + (K0 (α + r)/P0 - r) ] / (K0 (α + r))Therefore, P(t) becomes:P(t) = e^{r t} * (K0 (α + r)) / [ r e^{(α + r) t} + (K0 (α + r)/P0 - r) ]Simplify numerator and denominator:Numerator: K0 (α + r) e^{r t}Denominator: r e^{(α + r) t} + K0 (α + r)/P0 - rLet me write it as:P(t) = [ K0 (α + r) e^{r t} ] / [ r e^{(α + r) t} + K0 (α + r)/P0 - r ]This seems as simplified as it can get. Alternatively, we can factor out e^{r t} in the denominator:Denominator: r e^{(α + r) t} + K0 (α + r)/P0 - r = r e^{r t} e^{α t} + K0 (α + r)/P0 - rSo,P(t) = [ K0 (α + r) e^{r t} ] / [ r e^{r t} e^{α t} + K0 (α + r)/P0 - r ]Factor out e^{r t} in the first term:= [ K0 (α + r) e^{r t} ] / [ e^{r t} (r e^{α t}) + (K0 (α + r)/P0 - r) ]So,P(t) = [ K0 (α + r) ] / [ r e^{α t} + (K0 (α + r)/P0 - r) e^{-r t} ]Hmm, this seems similar to the previous expression. Maybe it's best to leave it in the form:P(t) = 1 / [ (r / (K0 (α + r))) e^{α t} + (1 / P0 - r / (K0 (α + r))) e^{-r t} ]Alternatively, we can write it as:P(t) = frac{1}{frac{r}{K_0 (alpha + r)} e^{alpha t} + left( frac{1}{P_0} - frac{r}{K_0 (alpha + r)} right) e^{-r t}}This is a valid expression, though it's a bit complex. I don't think it simplifies much further without additional constraints or specific values for the constants.So, summarizing, after solving the differential equation with the time-dependent carrying capacity K(t) = K0 e^{-α t}, the population P(t) is given by:P(t) = 1 / [ (r / (K0 (α + r))) e^{α t} + (1 / P0 - r / (K0 (α + r))) e^{-r t} ]I think that's the solution for part 2.Final Answer1. The population at time ( t ) is given by: [ boxed{P(t) = frac{K P_0 e^{rt}}{K + P_0 (e^{rt} - 1)}} ]2. With the time-dependent carrying capacity, the population becomes: [ boxed{P(t) = frac{1}{frac{r}{K_0 (alpha + r)} e^{alpha t} + left( frac{1}{P_0} - frac{r}{K_0 (alpha + r)} right) e^{-rt}}} ]</think>

🧠:<think>Alright, so I have this problem about Windsor-Tecumseh, Ontario, where a staunch left-wing voter is looking at demographic changes over the past decade to predict future voting outcomes. The problem has two main parts, both involving some calculations. Let me try to break them down step by step.First, the population growth is modeled by the function ( P(t) = 120,000 cdot e^{0.03t} ), where ( t ) is the number of years since 2010. I need to calculate the population in 2020 and 2030. Okay, so for 2020, that would be 10 years after 2010, so ( t = 10 ). Similarly, for 2030, that's 20 years after 2010, so ( t = 20 ). Let me write that down:For 2020:( P(10) = 120,000 cdot e^{0.03 times 10} )For 2030:( P(20) = 120,000 cdot e^{0.03 times 20} )I remember that ( e ) is approximately 2.71828, but I might need to use a calculator for the exponentials. Alternatively, I can compute the exponents first and then multiply by 120,000.Let me compute the exponent for 2020 first:0.03 * 10 = 0.3So, ( e^{0.3} ) is approximately... Hmm, I think ( e^{0.3} ) is about 1.34986. Let me verify that. Yes, because ( e^{0.3} ) is roughly 1.34986. So, multiplying that by 120,000:120,000 * 1.34986 ≈ 161,983.2So, approximately 161,983 people in 2020.Now, for 2030, the exponent is 0.03 * 20 = 0.6. So, ( e^{0.6} ) is approximately... I think it's around 1.82212. Let me check that. Yes, ( e^{0.6} ) is approximately 1.82212. So, multiplying that by 120,000:120,000 * 1.82212 ≈ 218,654.4So, approximately 218,654 people in 2030.Wait, let me make sure I did that correctly. Maybe I should use a calculator for more precise numbers, but since I don't have one handy, I'll go with these approximate values.Moving on to the second part. A survey in 2020 shows that 65% of the population supports left-wing policies, and this support is increasing at a continuous rate of 2% per year. I need to determine the expected percentage in 2030 and then estimate the number of left-wing supporters in 2030 using the population from the first part.So, the support rate is increasing continuously at 2% per year. That sounds like another exponential growth model, similar to the population growth. So, the formula should be similar: ( S(t) = S_0 cdot e^{rt} ), where ( S_0 ) is the initial support rate, ( r ) is the growth rate, and ( t ) is the time in years.In this case, ( S_0 ) is 65% in 2020, and the rate ( r ) is 2% per year, which is 0.02 in decimal. But wait, the problem says the support is increasing at a continuous rate of 2% per year. So, that would mean the growth rate is 0.02.But hold on, is the 2% per year a continuous rate or an annual rate? The problem says "continuous rate," so I think we can use the exponential model directly. So, yes, ( S(t) = 65% cdot e^{0.02t} ).But wait, let me clarify: if the support is increasing at a continuous rate of 2% per year, that would mean the differential equation ( dS/dt = 0.02S ), which leads to the exponential growth solution. So, yes, the formula is correct.But we need to find the support rate in 2030. Since the survey was conducted in 2020, the time ( t ) will be 10 years from 2020 to 2030. So, ( t = 10 ).Therefore, ( S(10) = 65% cdot e^{0.02 times 10} ).Calculating the exponent first: 0.02 * 10 = 0.2. So, ( e^{0.2} ) is approximately 1.22140. So, multiplying that by 65%:65% * 1.22140 ≈ 79.391%.So, approximately 79.39% of the population would support left-wing policies in 2030.Wait, that seems like a significant increase. Let me verify the calculation. 0.02 * 10 is 0.2, ( e^{0.2} ) is about 1.2214, so 65 * 1.2214 is indeed approximately 79.391. So, that seems correct.Now, to find the number of left-wing supporters in 2030, I need to take the percentage we just calculated and apply it to the population in 2030, which we found earlier to be approximately 218,654.So, the number of supporters would be 218,654 * (79.391 / 100).Calculating that: 218,654 * 0.79391 ≈ ?Let me compute that step by step. First, 200,000 * 0.79391 = 158,782.Then, 18,654 * 0.79391 ≈ Let's compute 18,654 * 0.7 = 13,057.8, and 18,654 * 0.09391 ≈ 1,753. So, adding them together: 13,057.8 + 1,753 ≈ 14,810.8.So, total supporters ≈ 158,782 + 14,810.8 ≈ 173,592.8.So, approximately 173,593 left-wing supporters in 2030.Wait, let me check that multiplication again because 218,654 * 0.79391 is a bit more precise.Alternatively, I can compute 218,654 * 0.79391 as follows:First, 218,654 * 0.7 = 153,057.8218,654 * 0.09 = 19,678.86218,654 * 0.00391 ≈ 218,654 * 0.004 = 874.616, subtract a bit: 874.616 - (218,654 * 0.00009) ≈ 874.616 - 19.678 ≈ 854.938So, adding them together: 153,057.8 + 19,678.86 = 172,736.66Then, adding 854.938: 172,736.66 + 854.938 ≈ 173,591.598So, approximately 173,592 supporters. So, my initial estimate was pretty close.Therefore, summarizing:1. Population in 2020: ~161,983Population in 2030: ~218,6542. Percentage supporting left-wing in 2030: ~79.39%Number of supporters: ~173,592Wait, but let me think again about the support rate calculation. The support is increasing at a continuous rate of 2% per year. So, the formula is correct, right? It's an exponential growth model with continuous compounding.Yes, because continuous growth is modeled by ( e^{rt} ). So, 65% growing at 2% per year continuously would indeed be 65 * e^{0.02t}.So, over 10 years, that's 65 * e^{0.2} ≈ 65 * 1.2214 ≈ 79.39%.So, that seems correct.Alternatively, if it were a simple annual increase of 2%, it would be 65 * (1.02)^10, which is approximately 65 * 1.21899 ≈ 79.234%, which is slightly less. But since the problem specifies a continuous rate, we should use the exponential function, so 79.39% is correct.Therefore, the number of supporters is approximately 173,592.Wait, but let me check the population in 2030 again. 120,000 * e^{0.03*20} = 120,000 * e^{0.6} ≈ 120,000 * 1.8221188 ≈ 218,654.26, which is about 218,654. So, that's correct.So, all in all, the calculations seem consistent.Just to recap:1. Population in 2020: ( P(10) = 120,000 * e^{0.3} ≈ 161,983 )Population in 2030: ( P(20) = 120,000 * e^{0.6} ≈ 218,654 )2. Support rate in 2030: ( S(10) = 65% * e^{0.2} ≈ 79.39% )Number of supporters: 218,654 * 0.7939 ≈ 173,592So, that should be the answer.Final Answer1. The population in 2020 is boxed{161983} and in 2030 is boxed{218654}.2. The expected percentage of left-wing supporters in 2030 is boxed{79.39%}, and the estimated number of left-wing supporters is boxed{173592}.</think>

🧠:<think>Okay, so I've got this problem about the Suez Crisis of 1956, and I need to figure out two things. First, calculate the average intensity of international diplomatic engagements using the given function E(t). Second, determine the decay constant λ for the economic impact model and compute I(t) at t=10 days. Let me break this down step by step.Starting with the first part: the function E(t) is given as 50 sin(πt/10) + 100. I need to find the average intensity over the entire period of the crisis. The crisis started on October 29, 1956, and ended on November 7, 1956. So, how many days is that? Let me count: from October 29 to November 7. October has 31 days, so from October 29 to October 31 is 3 days, and then November 1 to November 7 is 7 days. So total days are 3 + 7 = 10 days. So t ranges from 0 to 10 days.To find the average value of E(t) over this period, I remember that the average value of a function over an interval [a, b] is given by (1/(b-a)) times the integral of the function from a to b. So in this case, the average intensity, let's call it E_avg, would be (1/10) times the integral from t=0 to t=10 of E(t) dt.So E_avg = (1/10) ∫₀¹⁰ [50 sin(πt/10) + 100] dt.Let me compute this integral. I can split it into two parts: the integral of 50 sin(πt/10) dt plus the integral of 100 dt.First integral: ∫50 sin(πt/10) dt. The integral of sin(ax) dx is (-1/a) cos(ax) + C. So here, a is π/10, so the integral becomes 50 * (-10/π) cos(πt/10) + C. So that's (-500/π) cos(πt/10).Second integral: ∫100 dt is straightforward, it's 100t + C.So putting it all together, the integral from 0 to 10 is:[ (-500/π) cos(πt/10) + 100t ] evaluated from 0 to 10.Let me compute this at t=10:First term: (-500/π) cos(π*10/10) = (-500/π) cos(π) = (-500/π)*(-1) = 500/π.Second term: 100*10 = 1000.So total at t=10: 500/π + 1000.Now at t=0:First term: (-500/π) cos(0) = (-500/π)*1 = -500/π.Second term: 100*0 = 0.So total at t=0: -500/π.Subtracting t=0 from t=10:(500/π + 1000) - (-500/π) = 500/π + 1000 + 500/π = 1000 + 1000/π.So the integral is 1000 + 1000/π.Therefore, E_avg = (1/10)*(1000 + 1000/π) = (1000/10) + (1000/π)/10 = 100 + 100/π.Calculating that numerically, π is approximately 3.1416, so 100/π ≈ 31.83. So E_avg ≈ 100 + 31.83 ≈ 131.83.Wait, but let me double-check my integral calculation. The integral of 50 sin(πt/10) is indeed (-500/π) cos(πt/10). Evaluated from 0 to 10:At t=10: cos(π) = -1, so (-500/π)*(-1) = 500/π.At t=0: cos(0) = 1, so (-500/π)*(1) = -500/π.So the difference is 500/π - (-500/π) = 1000/π. Then the integral of 100 dt from 0 to10 is 1000. So total integral is 1000 + 1000/π. Then E_avg is (1000 + 1000/π)/10 = 100 + 100/π. So that's correct.So the average intensity is 100 + 100/π, which is approximately 131.83.Moving on to the second part: the economic impact I(t) is modeled by I(t) = I₀ e^{-λ t}. We're told that the impact was halved by the end of the crisis, which was at t=10 days. So I(10) = (1/2) I₀.So substituting into the equation: (1/2) I₀ = I₀ e^{-λ*10}.Divide both sides by I₀: 1/2 = e^{-10λ}.Take the natural logarithm of both sides: ln(1/2) = -10λ.We know that ln(1/2) is -ln(2), so -ln(2) = -10λ.Divide both sides by -10: λ = ln(2)/10.Calculating that, ln(2) is approximately 0.6931, so λ ≈ 0.6931/10 ≈ 0.06931 per day.Now, we need to compute I(t) for t=10 days. Wait, but wait, the crisis ended at t=10, and the impact was halved by then. So I(10) = (1/2) I₀, which we already know. But the question says "compute I(t) for t=10 days after the start of the crisis." So that would be t=10, which is exactly the end of the crisis. So I(10) = (1/2) I₀.But maybe they want it in terms of I₀? Or perhaps they want the value at t=10, which is 0.5 I₀.Alternatively, if they meant t=10 days after the start, which is the same as the end of the crisis, so yes, it's 0.5 I₀.Wait, but let me make sure. The crisis started on October 29, and t=10 would be November 8, right? Because October 29 is day 0, October 30 is day 1, ..., November 7 is day 10. So t=10 is November 7, which is the end of the crisis. So I(10) is indeed 0.5 I₀.But let me confirm: the problem says "the impact was halved by the end of the crisis (November 7, 1956)", which is t=10. So yes, I(10) = 0.5 I₀.So λ is ln(2)/10, approximately 0.06931 per day, and I(10) is 0.5 I₀.Wait, but the question says "compute I(t) for t=10 days after the start of the crisis." So that's exactly t=10, which is 0.5 I₀.So summarizing:1. The average intensity E_avg is 100 + 100/π, approximately 131.83.2. The decay constant λ is ln(2)/10, approximately 0.06931 per day, and I(10) is 0.5 I₀.I think that's it. Let me just make sure I didn't make any calculation errors.For the first part, the integral of sin over a full period would average out to zero, but since the period here is 20 days (since the period of sin(πt/10) is 20), but our interval is only 10 days, which is half a period. So the integral of sin over half a period would not be zero. Wait, let me think: from t=0 to t=10, which is half the period, the integral would be twice the integral from 0 to 10 of sin(πt/10) dt. Wait, no, the integral over half a period would actually be the same as the integral over a full period divided by two, but in this case, since it's a sine wave, the integral over half a period from 0 to π would be 2, but scaled appropriately.Wait, but in our case, the integral of sin(πt/10) from 0 to 10 is:Let me compute ∫₀¹⁰ sin(πt/10) dt.Let u = πt/10, so du = π/10 dt, so dt = 10/π du.When t=0, u=0; t=10, u=π.So ∫₀¹⁰ sin(πt/10) dt = ∫₀^π sin(u) * (10/π) du = (10/π) ∫₀^π sin(u) du = (10/π)[-cos(u)]₀^π = (10/π)[-cos(π) + cos(0)] = (10/π)[-(-1) + 1] = (10/π)(1 + 1) = 20/π.So the integral of sin(πt/10) from 0 to10 is 20/π. Therefore, the integral of 50 sin(πt/10) is 50*(20/π) = 1000/π.Then the integral of 100 dt from 0 to10 is 1000.So total integral is 1000 + 1000/π, which matches what I had before. So E_avg is (1000 + 1000/π)/10 = 100 + 100/π, which is correct.And for the second part, solving for λ when I(10) = 0.5 I₀ gives λ = ln(2)/10, which is correct.So I think my answers are correct.</think>