There are several ways we can define elasticity E. Each indicates how the quantity demanded changes as the price is changed. In the examples below, we’ll utilize elasticity defined as

$latex \displaystyle \text{E}\approx \frac{P}{Q}\,\frac{dQ}{dP}$

With this definition, the revenue is maximized when *E* = -1.

**Example 1** Suppose the demand function is defined by

$latex \displaystyle Q=30-\frac{P}{5}$

where *Q* units are demanded at a price of *P* dollars each.

a. Find an expression for the elasticity in terms of *P*.

b. Find the quantity at which the revenue is maximized.

**Example 2** Suppose the demand function is defined by

$latex \displaystyle Q=40-\frac{P}{4}$

where *Q* units are demanded at a price of *P* dollars each.

a. Find an expression for the elasticity in terms of *P*.

b. Find the quantity at which the revenue is maximized.

**Example 3** Suppose the demand function is defined by

$latex \displaystyle Q=39300-7{{P}^{2}}$

where *Q* units are demanded at a price of *P* dollars each.

a. Find an expression for the elasticity in terms of *P*.

b. Find the quantity at which the revenue is maximized.