1. **Problem Statement:** We are given the average cost function $$AC = \frac{100}{Q} + \frac{4}{\sqrt{Q}} + 3$$ and two points on the demand curve: $$A=\left(\frac{4}{3},1\right)$$ and $$B=\left(1,\frac{3}{2}\right)$$. We need to find: a) The total cost function. b) The total revenue function. c) The profit function. --- 2. **Find the total cost function:** The average cost function is defined as $$AC = \frac{C(Q)}{Q}$$ where $$C(Q)$$ is the total cost function. Rearranging, we get: $$C(Q) = AC \times Q = \left(\frac{100}{Q} + \frac{4}{\sqrt{Q}} + 3\right) Q$$ Simplify each term: - $$\frac{100}{Q} \times Q = 100$$ - $$\frac{4}{\sqrt{Q}} \times Q = 4Q^{1 - \frac{1}{2}} = 4Q^{\frac{1}{2}} = 4\sqrt{Q}$$ - $$3 \times Q = 3Q$$ So, $$C(Q) = 100 + 4\sqrt{Q} + 3Q$$ --- 3. **Find the total revenue function:** The demand function is linear and passes through points $$A\left(\frac{4}{3},1\right)$$ and $$B\left(1,\frac{3}{2}\right)$$ where the first coordinate is quantity $$Q$$ and the second is price $$P$$. Calculate the slope $$m$$: $$m = \frac{P_B - P_A}{Q_B - Q_A} = \frac{\frac{3}{2} - 1}{1 - \frac{4}{3}} = \frac{\frac{1}{2}}{-\frac{1}{3}} = -\frac{3}{2}$$ Use point-slope form with point $$A$$: $$P - 1 = -\frac{3}{2} \left(Q - \frac{4}{3}\right)$$ Simplify: $$P = 1 - \frac{3}{2}Q + 2 = 3 - \frac{3}{2}Q$$ So the demand (price) function is: $$P(Q) = 3 - \frac{3}{2}Q$$ Total revenue $$R(Q)$$ is price times quantity: $$R(Q) = P(Q) \times Q = \left(3 - \frac{3}{2}Q\right) Q = 3Q - \frac{3}{2}Q^2$$ --- 4. **Find the profit function:** Profit $$\pi(Q)$$ is total revenue minus total cost: $$\pi(Q) = R(Q) - C(Q) = \left(3Q - \frac{3}{2}Q^2\right) - \left(100 + 4\sqrt{Q} + 3Q\right)$$ Simplify: $$\pi(Q) = 3Q - \frac{3}{2}Q^2 - 100 - 4\sqrt{Q} - 3Q = -\frac{3}{2}Q^2 - 4\sqrt{Q} - 100$$ --- **Final answers:** - Total cost function: $$C(Q) = 100 + 4\sqrt{Q} + 3Q$$ - Total revenue function: $$R(Q) = 3Q - \frac{3}{2}Q^2$$ - Profit function: $$\pi(Q) = -\frac{3}{2}Q^2 - 4\sqrt{Q} - 100$$