1. **Problem Statement:** a) Find the market equilibrium price and quantity given demand equation $p = 3q - 22$ and supply equation $q^2 + 2p + 4q = 100$. b) Given marginal revenue function $MR = 25 - 3Q^2$, find the total revenue function. 2. **Step 1: Find Market Equilibrium (a)** Market equilibrium occurs where quantity demanded equals quantity supplied, so demand price equals supply price. Demand: $p = 3q - 22$ Supply: $q^2 + 2p + 4q = 100$ Solve supply for $p$: $$2p = 100 - q^2 - 4q$$ $$p = \frac{100 - q^2 - 4q}{2}$$ 3. **Step 2: Set demand price equal to supply price:** $$3q - 22 = \frac{100 - q^2 - 4q}{2}$$ Multiply both sides by 2: $$2(3q - 22) = 100 - q^2 - 4q$$ $$6q - 44 = 100 - q^2 - 4q$$ 4. **Step 3: Rearrange to form quadratic equation:** $$6q - 44 - 100 + q^2 + 4q = 0$$ $$q^2 + 10q - 144 = 0$$ 5. **Step 4: Solve quadratic equation:** Use quadratic formula $q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=10$, $c=-144$. $$\Delta = 10^2 - 4(1)(-144) = 100 + 576 = 676$$ $$q = \frac{-10 \pm \sqrt{676}}{2} = \frac{-10 \pm 26}{2}$$ Two solutions: $$q_1 = \frac{-10 + 26}{2} = 8$$ $$q_2 = \frac{-10 - 26}{2} = -18$$ (discard negative quantity) 6. **Step 5: Find equilibrium price $p$ using demand equation:** $$p = 3(8) - 22 = 24 - 22 = 2$$ 7. **Step 6: Calculate total revenue $TR = p \times q$:** $$TR = 2 \times 8 = 16$$ 8. **Step 7: Find total revenue function from marginal revenue (b):** Given $MR = \frac{dTR}{dQ} = 25 - 3Q^2$ Integrate $MR$ with respect to $Q$: $$TR = \int (25 - 3Q^2) dQ = 25Q - Q^3 + C$$ Assuming $TR = 0$ when $Q=0$, constant $C=0$. So, $$TR = 25Q - Q^3$$ **Final answers:** - Market equilibrium quantity $q = 8$ - Market equilibrium price $p = 2$ - Total revenue at equilibrium $TR = 16$ - Total revenue function $TR = 25Q - Q^3$