1. **Problem statement:** Given the demand function for a monopolist $Q = 150 - 4P$, find expressions for Total Revenue (TR), Marginal Revenue (MR), and Average Revenue (AR). Then evaluate TR, MR, and AR at $Q=10$ and explain the meaning of these values. 2. **Step 1: Express price $P$ as a function of quantity $Q$.** From the demand function: $$Q = 150 - 4P$$ Rearranging for $P$: $$4P = 150 - Q$$ $$P = \frac{150 - Q}{4} = 37.5 - 0.25Q$$ 3. **Step 2: Find Total Revenue (TR).** Total Revenue is price times quantity: $$TR = P \times Q = \left(37.5 - 0.25Q\right)Q = 37.5Q - 0.25Q^2$$ 4. **Step 3: Find Marginal Revenue (MR).** Marginal Revenue is the derivative of Total Revenue with respect to quantity: $$MR = \frac{dTR}{dQ} = \frac{d}{dQ}(37.5Q - 0.25Q^2) = 37.5 - 0.5Q$$ 5. **Step 4: Find Average Revenue (AR).** Average Revenue is Total Revenue divided by quantity, which equals price: $$AR = \frac{TR}{Q} = \frac{37.5Q - 0.25Q^2}{Q} = 37.5 - 0.25Q$$ 6. **Step 5: Evaluate TR, MR, and AR at $Q=10$.** $$TR(10) = 37.5(10) - 0.25(10)^2 = 375 - 25 = 350$$ $$MR(10) = 37.5 - 0.5(10) = 37.5 - 5 = 32.5$$ $$AR(10) = 37.5 - 0.25(10) = 37.5 - 2.5 = 35$$ 7. **Step 6: Interpretation.** - $TR(10) = 350$ means total revenue from selling 10 units is 350. - $MR(10) = 32.5$ means selling one more unit beyond 10 increases revenue by 32.5. - $AR(10) = 35$ means average revenue (price) per unit at 10 units sold is 35. Since $MR(10) < AR(10)$, the revenue gained from selling an additional unit is less than the average revenue per unit, reflecting the downward sloping demand curve. This means to sell more units, the monopolist must lower the price, reducing revenue gained on all units sold.