Problem

A pizza restaurant sells an average of 80 pizzas per day at its usual price of $12.95. It experiments with sales and coupons for dollars off the usual price, and finds that the number of pizzas sold when the price decreases by 2 dollars is 135. It estimates that the number of pizzas sold when the price goes down by x dollars is modeled by the function 50 ln(x + 1) + 80.

Use linear approximation to find the change in the number of pizzas sold when the price drops from $10.95 to $9.95.

Solution

A drop from $12.95 to $10.95 amounts to 2 dollars off the regular price. A linear approximation to f(x) = 50 ln(x + 1) + 80 around a = 2 is of the form

We are given f(2) = 135, and we need to calculate f '(2).

The linear approximation to the number of pizzas is

A price drop from $10.95 to $9.95 means x = 3. The approximated number of pizzas the restaurant will sell is

or about 152 pizzas.