A product sells according to the demand function \( q=16 p-p^{2} \). Find the price elasticity of demand when the selling price is \( \$ 11.51 \). The price elasticity of demand is increase revenue, the price must ? when the price is \( \$ 11.51 \) dollars. To

To determine the **price elasticity of demand** and the appropriate pricing strategy to **increase revenue** when the selling price is **\$11.51**, follow these steps: ### 1. **Demand Function** Given the demand function: \[ q = 16p - p^2 \] where: - \( q \) = quantity demanded - \( p \) = price ### 2. **Calculate the Derivative \( \frac{dq}{dp} \)** \[ \frac{dq}{dp} = \frac{d}{dp}(16p - p^2) = 16 - 2p \] At \( p = \$11.51 \): \[ \frac{dq}{dp} = 16 - 2(11.51) = 16 - 23.02 = -7.02 \] ### 3. **Determine Quantity Demanded \( q \) at \( p = \$11.51 \)** \[ q = 16(11.51) - (11.51)^2 = 184.16 - 132.48 = 51.68 \] ### 4. **Calculate Price Elasticity of Demand \( E \)** The price elasticity of demand is given by: \[ E = \frac{dq}{dp} \times \frac{p}{q} \] Plugging in the values: \[ E = (-7.02) \times \frac{11.51}{51.68} \approx -7.02 \times 0.2226 \approx -1.566 \] ### 5. **Interpretation of Elasticity** - **Elasticity Value**: \( E \approx -1.566 \) - **Absolute Value**: \( |E| = 1.566 \) (greater than 1) Since \( |E| > 1 \), the **demand is elastic** at this price point. ### 6. **Pricing Strategy to Increase Revenue** - **Elastic Demand (\( |E| > 1 \))**: Decreasing the price will increase total revenue. - **Reasoning**: When demand is elastic, consumers are sensitive to price changes. Lowering the price leads to a proportionally larger increase in quantity demanded, thereby increasing total revenue. ### **Conclusion** - **Price Elasticity of Demand at \$11.51**: Approximately **-1.566**. - **Recommended Action to Increase Revenue**: **Decrease the price** from **\$11.51**.