To find the consumer surplus given the demand function \( P = \frac{100}{Q + 2} \) and the market price \( P = 20 \), follow these steps: 1. **Find the quantity \( Q \) at the market price \( P = 20 \)**: \[ 20 = \frac{100}{Q + 2} \] Solve for \( Q \): \[ 20(Q + 2) = 100 \] \[ 20Q + 40 = 100 \] \[ 20Q = 60 \] \[ Q = 3 \] 2. **Set up the integral for consumer surplus**: Consumer surplus is the area between the demand curve and the market price, from \( Q = 0 \) to \( Q = 3 \): \[ \text{Consumer Surplus} = \int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ \] 3. **Evaluate the integral**: \[ \int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ = \int_0^3 \frac{100}{Q + 2} \, dQ - \int_0^3 20 \, dQ \] 4. **Find the antiderivatives**: \[ \int \frac{100}{Q + 2} \, dQ = 100 \ln|Q + 2| + C \] \[ \int 20 \, dQ = 20Q + C \] 5. **Apply the limits of integration**: \[ \left[ 100 \ln|Q + 2| \right]_0^3 - \left[ 20Q \right]_0^3 \] 6. **Evaluate the definite integrals**: \[ \left[ 100 \ln|Q + 2| \right]_0^3 = 100 \ln(5) - 100 \ln(2) \] \[ \left[ 20Q \right]_0^3 = 20(3) - 20(0) = 60 \] 7. **Combine the results**: \[ 100 \ln(5) - 100 \ln(2) - 60 \] \[ 100 (\ln(5) - \ln(2)) - 60 \] \[ 100 \ln\left(\frac{5}{2}\right) - 60 \] 8. **Calculate the numerical value**: \[ 100 \ln\left(\frac{5}{2}\right) - 60 \approx 100 \times 0.9163 - 60 \approx 91.63 - 60 \approx 31.63 \] So, the consumer surplus is approximately \( 31.6291 \). The correct answer is: - \( 31.6291 \)

To find the consumer surplus given the demand function $P = \frac{100}{Q + 2}$ and the market price $P = 20$, follow these steps: 1. Find the quantity $Q$ at the market price $P = 20$: $20 = \frac{100}{Q + 2}$ Solve for $Q$: $20(Q + 2) = 100$ $20Q + 40 = 100$ $20Q = 60$ $Q = 3$ 2. Set up the integral for consumer surplus: Consumer surplus is the area between the demand curve and the market price, from $Q = 0$ to $Q = 3$: $\text{Consumer Surplus} = \int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ$ 3. Evaluate the integral: $\int_0^3 \left( \frac{100}{Q + 2} - 20 \right) \, dQ = \int_0^3 \frac{100}{Q + 2} \, dQ - \int_0^3 20 \, dQ$ 4. Find the antiderivatives: $\int \frac{100}{Q + 2} \, dQ = 100 \ln|Q + 2| + C$ $\int 20 \, dQ = 20Q + C$ 5. Apply the limits of integration: $\left[ 100 \ln|Q + 2| \right]_0^3 - \left[ 20Q \right]_0^3$ 6. Evaluate the definite integrals: $\left[ 100 \ln|Q + 2| \right]_0^3 = 100 \ln(5) - 100 \ln(2)$ $\left[ 20Q \right]_0^3 = 20(3) - 20(0) = 60$ 7. Combine the results: $100 \ln(5) - 100 \ln(2) - 60$ $100 (\ln(5) - \ln(2)) - 60$ $100 \ln\left(\frac{5}{2}\right) - 60$ 8. Calculate the numerical value: $100 \ln\left(\frac{5}{2}\right) - 60 \approx 100 \times 0.9163 - 60 \approx 91.63 - 60 \approx 31.63$ So, the consumer surplus is approximately $31.6291$. The correct answer is: - $31.6291$