(a) Find the formula for the elasticity of demand at any price as a function of the variable p . E(p)

To find the elasticity of demand $E(p)$ at any price $p$, we can use the formula for price elasticity of demand, which is defined as:

$E(p) = \frac{dq/dp \cdot p}{q}$

1. Find $q$: From the demand function, we have: $q = 400 - p^2$

2. Find $\frac{dq}{dp}$: Differentiate $q$ with respect to $p$: $\frac{dq}{dp} = -2p$

3. Substitute $q$ and $\frac{dq}{dp}$ into the elasticity formula: $E(p) = \frac{(-2p) \cdot p}{400 - p^2}$ Simplifying this gives: $E(p) = \frac{-2p^2}{400 - p^2}$

Thus, the formula for the elasticity of demand at any price $p$ is:

$E(p) = \frac{-2p^2}{400 - p^2}$

Would you like further details or explanations? Here are some related questions to consider:

1. How does the elasticity of demand change as $p$ increases?
2. What are the implications of elasticity being negative?
3. How can we determine the point of unit elasticity?
4. How does this demand function compare to linear demand functions?
5. What factors could affect the demand function in real-world scenarios?

Tip: Remember that elasticity values greater than 1 indicate elastic demand, while values less than 1 indicate inelastic demand.