1. **Problem 1: Calculate the consumers' surplus under monopoly given the demand function.** Since the demand function is not explicitly provided, let's assume a general linear demand function of the form $$P = a - bQ$$ where $P$ is price and $Q$ is quantity. **Step 1:** Recall that consumers' surplus (CS) is the area between the demand curve and the market price line, up to the quantity sold. **Step 2:** The formula for consumers' surplus is: $$CS = \int_0^{Q^*} P(Q) \, dQ - P^* Q^*$$ where $Q^*$ is the equilibrium quantity and $P^*$ is the equilibrium price. **Step 3:** For a monopolist, $Q^*$ and $P^*$ are determined by maximizing profit, which involves equating marginal revenue (MR) to marginal cost (MC). Without explicit functions or values, we cannot compute numerical answers. **Step 4:** If you provide the demand function and cost details, I can compute the exact consumers' surplus. --- 2. **Problem 2: Given the rate of investment $I(t) = 9t^{1/2}$, find the level of capital formation.** (a) **Level of capital formation in 8 years:** **Step 1:** Capital formation $K(t)$ is the integral of the investment rate over time: $$K(t) = \int_0^t I(x) \, dx = \int_0^t 9x^{1/2} \, dx$$ **Step 2:** Compute the integral: $$K(t) = 9 \int_0^t x^{1/2} \, dx = 9 \left[ \frac{2}{3} x^{3/2} \right]_0^t = 9 \times \frac{2}{3} t^{3/2} = 6 t^{3/2}$$ **Step 3:** Substitute $t=8$: $$K(8) = 6 \times 8^{3/2} = 6 \times (8^{1/2})^3 = 6 \times (2.8284)^3 = 6 \times 22.6274 = 135.7644$$ So, the level of capital formation after 8 years is approximately $135.76$ units. (b) **Level of capital formation from year 5 through year 8:** **Step 1:** Calculate $K(8)$ and $K(5)$: $$K(5) = 6 \times 5^{3/2} = 6 \times (\sqrt{5})^3 = 6 \times (2.2361)^3 = 6 \times 11.1803 = 67.0818$$ **Step 2:** The capital formed between year 5 and 8 is: $$K(8) - K(5) = 135.7644 - 67.0818 = 68.6826$$ So, the capital formation from year 5 through 8 is approximately $68.68$ units. --- **Final answers:** 1. Consumers' surplus requires the explicit demand function and cost details to compute. 2. (a) Capital formation in 8 years: approximately $135.76$ (b) Capital formation from year 5 to 8: approximately $68.68$