1. **State the problem:** We have demand and supply functions: $$q_D(p) = 525 - 2\sqrt{p}$$ $$q_S(p) = 45 + 2p$$ We want to find the equilibrium price $p^*$ and quantity $q^*$ where $q_D = q_S$, and then calculate the profit at equilibrium. 2. **Set equilibrium condition:** At equilibrium, quantity demanded equals quantity supplied: $$525 - 2\sqrt{p} = 45 + 2p$$ 3. **Rearrange the equation:** $$525 - 45 = 2p + 2\sqrt{p}$$ $$480 = 2p + 2\sqrt{p}$$ Divide both sides by 2: $$\cancel{2} \times 240 = \cancel{2}p + \cancel{2}\sqrt{p}$$ $$240 = p + \sqrt{p}$$ 4. **Substitute $x = \sqrt{p}$, so $p = x^2$:** $$240 = x^2 + x$$ Rearranged: $$x^2 + x - 240 = 0$$ 5. **Solve quadratic equation:** Using quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=1$, $c=-240$: $$x = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-240)}}{2} = \frac{-1 \pm \sqrt{1 + 960}}{2} = \frac{-1 \pm \sqrt{961}}{2}$$ $$\sqrt{961} = 31$$ So, $$x = \frac{-1 \pm 31}{2}$$ 6. **Select positive root (since $x=\sqrt{p} \geq 0$):** $$x = \frac{-1 + 31}{2} = \frac{30}{2} = 15$$ 7. **Find equilibrium price:** $$p^* = x^2 = 15^2 = 225$$ 8. **Find equilibrium quantity:** Use supply function: $$q^* = 45 + 2p^* = 45 + 2 \times 225 = 45 + 450 = 495$$ 9. **Calculate profit:** Fixed costs = 80000 Variable cost per unit = 1.60 Total cost = Fixed cost + Variable cost \times quantity $$C = 80000 + 1.60 \times 495 = 80000 + 792 = 80792$$ Total revenue = price \times quantity $$R = 225 \times 495 = 111375$$ Profit = Revenue - Cost $$\pi = 111375 - 80792 = 30583$$ **Final answers:** - Equilibrium price: $p^* = 225$ - Equilibrium quantity: $q^* = 495$ - Profit at equilibrium: $30583$