For the demand function D(p), complete the following.
D(p) = 2700 − p^2
(a) Find the elasticity of demand E(p).
E(p) = ________

To find the elasticity of demand \( E(p) \) for the demand function \( D(p) = 2700 - p^2 \), we use the formula for the price elasticity of demand:

\[ E(p) = \left| \frac{dD(p)}{dp} \cdot \frac{p}{D(p)} \right| \]

First, we find the derivative of \( D(p) \) with respect to \( p \):

\[ D(p) = 2700 - p^2 \]
\[ \frac{dD(p)}{dp} = -2p \]

Next, substitute \( \frac{dD(p)}{dp} \), \( p \), and \( D(p) \) into the elasticity formula:

\[ E(p) = \left| \frac{-2p \cdot p}{2700 - p^2} \right| = \left| \frac{-2p^2}{2700 - p^2} \right| \]

Therefore, the elasticity of demand \( E(p) \) is:

\[ E(p) = \left| \frac{2p^2}{2700 - p^2} \right| \]

This is the formula for the elasticity of demand for the given demand function.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What does it mean for a demand function to be elastic or inelastic?
2. How does the elasticity of demand change with respect to different price levels \( p \)?
3. Calculate the elasticity of demand when \( p = 30 \).
4. What are the implications of a perfectly elastic or inelastic demand?
5. How can businesses use the elasticity of demand to set their pricing strategy?
6. What are other forms of elasticity in economics besides price elasticity of demand?
7. How does the concept of elasticity relate to consumer behavior?
8. Explain the significance of the elasticity of demand being greater than 1, less than 1, or equal to 1.

**Tip:** When calculating elasticity, always consider the absolute value since elasticity is a measure of responsiveness and should be non-negative.

Learn how to calculate the elasticity of demand for the function D(p) = 2700 - p^2 using the elasticity formula. This problem involves derivatives and the concept of elasticity in economics. Suitable for college-level students studying economics or mathematics.

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