1. **State the problem:** We are given the supply curve $p = \sqrt{9 + x}$ and the quantity sold $x = 7$ units. We need to find the Producer's Surplus. 2. **Recall the formula for Producer's Surplus:** Producer's Surplus = Total Revenue - Variable Cost Alternatively, it can be calculated as the area above the supply curve and below the market price up to the quantity sold: $$\text{Producer's Surplus} = p(x) \times x - \int_0^x S(t) \, dt$$ where $S(t)$ is the supply function and $p(x)$ is the price at quantity $x$. 3. **Calculate the price at $x=7$:** $$p(7) = \sqrt{9 + 7} = \sqrt{16} = 4$$ 4. **Calculate total revenue:** $$TR = p(7) \times 7 = 4 \times 7 = 28$$ 5. **Calculate the integral of the supply curve from 0 to 7:** $$\int_0^7 \sqrt{9 + t} \, dt$$ Let $u = 9 + t$, then $du = dt$. When $t=0$, $u=9$; when $t=7$, $u=16$. So, $$\int_9^{16} \sqrt{u} \, du = \int_9^{16} u^{1/2} \, du = \left[ \frac{2}{3} u^{3/2} \right]_9^{16} = \frac{2}{3} (16^{3/2} - 9^{3/2})$$ Calculate powers: $$16^{3/2} = (\sqrt{16})^3 = 4^3 = 64$$ $$9^{3/2} = (\sqrt{9})^3 = 3^3 = 27$$ So, $$\int_0^7 \sqrt{9 + t} \, dt = \frac{2}{3} (64 - 27) = \frac{2}{3} \times 37 = \frac{74}{3} \approx 24.67$$ 6. **Calculate Producer's Surplus:** $$PS = TR - \text{Area under supply curve} = 28 - \frac{74}{3} = \frac{84}{3} - \frac{74}{3} = \frac{10}{3} \approx 3.33$$ **Final answer:** The Producer's Surplus is $$\frac{10}{3}$$ or approximately 3.33 units.