Let's solve the problem step by step. --- ### **Given:** **Student demand:** \[ Q^S = 1,760 - 100P \] **Adult demand:** \[ Q^A = 2,000 - 100P \] **Cost:** \[ C(Q) = 1Q + 0.005Q^2 \] where \( Q = Q^S + Q^A \) --- ## **a. Prices and Profit with Price Discrimination** ### **Step 1: Find MR for Each Group** #### **For Students:** Inverse demand: \[ Q^S = 1,760 - 100P \implies P = 17.6 - 0.01Q^S \] Revenue: \[ R^S = P \cdot Q^S = (17.6 - 0.01Q^S)Q^S = 17.6Q^S - 0.01(Q^S)^2 \] Marginal Revenue: \[ MR^S = \frac{dR^S}{dQ^S} = 17.6 - 0.02Q^S \] #### **For Adults:** Inverse demand: \[ Q^A = 2,000 - 100P \implies P = 20 - 0.01Q^A \] Revenue: \[ R^A = (20 - 0.01Q^A)Q^A = 20Q^A - 0.01(Q^A)^2 \] Marginal Revenue: \[ MR^A = 20 - 0.02Q^A \] --- ### **Step 2: Marginal Cost** Total Output: \( Q = Q^S + Q^A \) Cost: \( C(Q) = 1Q + 0.005Q^2 \) Marginal Cost: \[ MC = \frac{dC}{dQ} = 1 + 0.01Q \] --- ### **Step 3: Equate MR to MC for Each Group** #### **Students:** \[ MR^S = MC \implies 17.6 - 0.02Q^S = 1 + 0.01(Q^S + Q^A) \] #### **Adults:** \[ MR^A = MC \implies 20 - 0.02Q^A = 1 + 0.01(Q^S + Q^A) \] --- ### **Step 4: Solve the System** Let \( x = Q^S \), \( y = Q^A \), \( Q = x + y \): **Students:** \[ 17.6 - 0.02x = 1 + 0.01(x + y) \implies 17.6 - 1 = 0.02x + 0.01x + 0.01y \implies 16.6 = 0.03x + 0.01y \] **Adults:** \[ 20 - 0.02y = 1 + 0.01(x + y) \implies 19 = 0.01x + 0.03y \] Now solve: \[ \begin{cases} 16.6 = 0.03x + 0.01y \\ 19 = 0.01x + 0.03y \\ \end{cases} \] Multiply first by 3: \( 49.8 = 0.09x + 0.03y \) Subtract the second: \[ 49.8 - 19 = 0.09x + 0.03y - (0.01x + 0.03y) \implies 30.8 = 0.08x \implies x = \frac{30.8}{0.08} = 385 \] Plug \( x = 385 \) into the second: \[ 19 = 0.01(385) + 0.03y \implies 19 = 3.85 + 0.03y \implies 15.15 = 0.03y \implies y = \frac{15.15}{0.03} = 505 \] --- ### **Step 5: Find Prices** - **Student Price:** \( Q^S = 385 \) \( P^S = 17.6 - 0.01(385) = 17.6 - 3.85 = \boxed{13.75} \) - **Adult Price:** \( Q^A = 505 \) \( P^A = 20 - 0.01(505) = 20 - 5.05 = \boxed{14.95} \) --- ### **Step 6: Profit** Total output \( Q = 385 + 505 = 890 \) - **Total Revenue:** \( TR = P^S \times Q^S + P^A \times Q^A \) \( = 13.75 \times 385 + 14.95 \times 505 \) \( = 5293.75 + 7549.75 = 12843.50 \) - **Total Cost:** \( C(Q) = 1Q + 0.005Q^2 = 1 \times 890 + 0.005 \times 890^2 \) \( = 890 + 0.005 \times 792100 \) \( = 890 + 3960.5 = 4850.50 \) - **Profit:** \( \text{Profit} = 12843.50 - 4850.50 = \boxed{7993.00} \) --- ## **b. If Adult demand increases to \( Q^A = 2,200 - 100P \):** Repeat Steps 1–6 with new adult demand. **Inverse demand:** \( Q^A = 2,200 - 100P \implies P = 22 - 0.01Q^A \) \( R^A = 22Q^A - 0.01(Q^A)^2 \) \( MR^A = 22 - 0.02Q^A \) Set up the new system: - For students: \( 17.6 - 0.02x = 1 + 0.01(x + y) \implies 16.6 = 0.03x + 0.01y \) - For adults: \( 22 - 0.02y = 1 + 0.01(x + y) \implies 21 = 0.01x + 0.03y \) Multiply first by 3: \( 49.8 = 0.09x + 0.03y \) Subtract the second: \[ 49.8 - 21 = (0.09x + 0.03y) - (0.01x + 0.03y) \] \[ 28.8 = 0.08x \implies x = \frac{28.8}{0.08} = 360 \] Plug \( x = 360 \) into second: \[ 21 = 0.01(360) + 0.03y \implies 21 = 3.6 + 0.03y \implies 17.4 = 0.03y \implies y = \frac{17.4}{0.03} = 580 \] Now, calculate prices: - Student: \( P^S = 17.6 - 0.01(360) = 17.6 - 3.60 = \boxed{14.00} \) - Adult: \( P^A = 22 - 0.01(580) = 22 - 5.80 = \boxed{16.20} \) Calculate profit: - \( Q = 360 + 580 = 940 \) - \( TR = 14.00 \times 360 + 16.20 \times 580 = 5040 + 9396 = 14436 \) - \( TC = 1 \times 940 + 0.005 \times 940^2 = 940 + 0.005 \times 883600 = 940 + 4418 = 5358 \) - \( \text{Profit} = 14436 - 5358 = \boxed{9078} \) --- ## **c. When adult demand increases, the adult price:** **Increases** (from $14.95 to $16.20). --- ## **d. When adult demand increases, the student price:** **Increases** (from $13.75 to $14.00). --- ## **Final Answers (rounded to 2 decimal places):** ### a. - \( P_{student} = \$13.75 \) - \( P_{adult} = \$14.95 \) - Profit = \$7,993.00 ### b. - \( P_{student} = \$14.00 \) - \( P_{adult} = \$16.20 \) - Profit = \$9,078.00 ### c. - **Adult price** increases. ### d. - **Student price** increases. --- **Let me know if you need the boxed values filled in or further explanation!**