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We are given the demand function $ q = 12p - p^2 $, and we are asked to find the price elasticity of demand when the selling price is $ p = 11.41 $.

### Steps to solve:
1. **Elasticity of Demand Formula**: The price elasticity of demand $ E(p) $ is given by:
   \[
   E(p) = \left( \frac{p}{q} \right) \cdot \frac{dq}{dp}
   \]
   where $ q $ is the quantity demanded as a function of price $ p $, and $ \frac{dq}{dp} $ is the derivative of the demand function with respect to price.

2. **Find the derivative of the demand function**: 
   The demand function is $ q = 12p - p^2 $. We take its derivative with respect to $ p $:
   \[
   \frac{dq}{dp} = 12 - 2p
   \]

3. **Evaluate at $ p = 11.41 $**:
   \[
   \frac{dq}{dp} = 12 - 2(11.41) = 12 - 22.82 = -10.82
   \]

4. **Find the quantity $ q $ at $ p = 11.41 $**:
   Substitute $ p = 11.41 $ into the demand function:
   \[
   q = 12(11.41) - (11.41)^2 = 136.92 - 130.14 = 6.78
   \]

5. **Substitute values into the elasticity formula**:
   Now, we use the elasticity formula:
   \[
   E(11.41) = \left( \frac{11.41}{6.78} \right) \cdot (-10.82) = 1.682 \cdot (-10.82) \approx -18.2
   \]

The price elasticity of demand when the price is $ 11.41 $ is approximately $ -18.2 $, indicating that the demand is highly elastic at this price.

#### Answer:
- The price elasticity of demand is approximately $ -18.2 $.
- Since the price elasticity is negative and its absolute value is much greater than 1, it means the demand is **very elastic**. To increase revenue, the price should be **increased**, since a small price increase will not significantly affect the quantity demanded.

Would you like more details or have any other questions?

### Related questions:
1. How does the elasticity of demand change at different price levels?
2. What does it mean when the elasticity of demand is elastic, inelastic, or unitary?
3. How is revenue related to price elasticity of demand?
4. What factors affect the elasticity of demand for a product?
5. How does the demand function impact pricing strategies for maximizing revenue?

### Tip:
When the price elasticity of demand is greater than 1 (in absolute value), demand is elastic, and lowering prices typically increases total revenue, while increasing prices decreases it.

A product sells according to the demand function q = 12p - p^2. Find the price elasticity of demand when the selling price is $11.41.

Learn how to calculate the price elasticity of demand when the demand function is q = 12p - p^2 and the price is $11.41. This problem involves finding the derivative of the demand function and using it to calculate elasticity.

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