1. **Problem statement:** A firm produces quantity $q \geq 0$ with inverse demand $p(q) = 30 - q$ and total cost $C(q) = 50 + 4q + \frac{1}{2}q^2$. 2. **Compute average cost $AC(q)$ and marginal cost $MC(q)$:** - Average cost is $AC(q) = \frac{C(q)}{q}$ for $q > 0$. - Marginal cost is the derivative $MC(q) = C'(q)$. 3. **Calculate $AC(q)$:** $$AC(q) = \frac{50 + 4q + \frac{1}{2}q^2}{q} = \frac{50}{q} + 4 + \frac{1}{2}q$$ 4. **Calculate $MC(q)$:** $$MC(q) = \frac{d}{dq} \left(50 + 4q + \frac{1}{2}q^2\right) = 0 + 4 + q = 4 + q$$ 5. **Compute revenue $R(q)$ and profit $\Pi(q)$:** - Revenue is price times quantity: $R(q) = p(q) \times q = (30 - q)q = 30q - q^2$ - Profit is revenue minus cost: $$\Pi(q) = R(q) - C(q) = (30q - q^2) - \left(50 + 4q + \frac{1}{2}q^2\right) = 30q - q^2 - 50 - 4q - \frac{1}{2}q^2$$ 6. **Simplify profit:** $$\Pi(q) = (30q - 4q) - \left(q^2 + \frac{1}{2}q^2\right) - 50 = 26q - \frac{3}{2}q^2 - 50$$ 7. **Find break-even quantities where $\Pi(q) = 0$:** $$26q - \frac{3}{2}q^2 - 50 = 0$$ Multiply both sides by 2 to clear fraction: $$2 \times 26q - 2 \times \frac{3}{2}q^2 - 2 \times 50 = 0 \Rightarrow 52q - 3q^2 - 100 = 0$$ Rewrite: $$-3q^2 + 52q - 100 = 0$$ Multiply both sides by $\cancel{-1}$: $$\cancel{-1} \times (-3q^2 + 52q - 100) = 3q^2 - 52q + 100 = 0$$ 8. **Solve quadratic $3q^2 - 52q + 100 = 0$ using quadratic formula:** $$q = \frac{52 \pm \sqrt{(-52)^2 - 4 \times 3 \times 100}}{2 \times 3} = \frac{52 \pm \sqrt{2704 - 1200}}{6} = \frac{52 \pm \sqrt{1504}}{6}$$ 9. **Calculate $\sqrt{1504}$:** $$\sqrt{1504} \approx 38.78$$ 10. **Find roots:** $$q_1 = \frac{52 - 38.78}{6} \approx \frac{13.22}{6} \approx 2.20$$ $$q_2 = \frac{52 + 38.78}{6} \approx \frac{90.78}{6} \approx 15.13$$ 11. **Determine profit-maximizing quantity $q^*$:** Maximize $\Pi(q) = 26q - \frac{3}{2}q^2 - 50$. Set derivative $\Pi'(q) = 0$: $$\Pi'(q) = 26 - 3q = 0 \Rightarrow 3q = 26 \Rightarrow q^* = \frac{26}{3} \approx 8.67$$ 12. **Find corresponding price $p(q^*)$:** $$p(q^*) = 30 - q^* = 30 - 8.67 = 21.33$$ 13. **Interpretation:** The optimal output $q^* \approx 8.67$ lies between the break-even quantities $2.20$ and $15.13$. This means the firm produces at a level where profit is positive, above the lower break-even point and below the higher one. **Final answers:** - $AC(q) = \frac{50}{q} + 4 + \frac{1}{2}q$ - $MC(q) = 4 + q$ - $R(q) = 30q - q^2$ - $\Pi(q) = 26q - \frac{3}{2}q^2 - 50$ - Break-even quantities: $q \approx 2.20$ and $q \approx 15.13$ - Profit-maximizing quantity: $q^* = \frac{26}{3} \approx 8.67$ - Corresponding price: $p(q^*) \approx 21.33$