Math Problem Statement

In this problem, p is in dollars and q is the number of units.

The demand function for a product is given by q = 400 − p2 .

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

Solution

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

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

  1. Find qq: From the demand function, we have: q=400p2q = 400 - p^2

  2. Find dqdp\frac{dq}{dp}: Differentiate qq with respect to pp: dqdp=2p\frac{dq}{dp} = -2p

  3. Substitute qq and dqdp\frac{dq}{dp} into the elasticity formula: E(p)=(2p)p400p2E(p) = \frac{(-2p) \cdot p}{400 - p^2} Simplifying this gives: E(p)=2p2400p2E(p) = \frac{-2p^2}{400 - p^2}

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

E(p)=2p2400p2E(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 pp 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.

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Math Problem Analysis

Mathematical Concepts

Calculus
Elasticity of Demand
Functions

Formulas

E(p) = (dq/dp * p) / q
q = 400 - p^2
dq/dp = -2p

Theorems

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Suitable Grade Level

Grades 11-12