Calculus Word Problem

Question

Bob sells tires. He knows that his weekly sales drop if he increases the unit price p (in dollars). The weekly sales are given by an unknown function N(p). His weekly revenue is R(p)=pN(p).

Bob is currently selling tires for 55 dollars each. His weekly sales are running at 140 tires per week, so

N(55)=140.

His marketing department estimates that he will lose 2 sales per week for each 10 dollar increase in unit price, N′(55)=−0.2.

Estimate Bob's increase in weekly revenue for each one dollar increase in tire price.

Revenue increase =  ? dollars

Answer 1

To find the revenue increase estimate, you need to find R'(p). Since R(p)=pN(p), we can use product rule to find this derivative at p=55:

R'(p) = pN'(p) + p'N(p)

R'(55) = 55•(-0.2) + 1•140

= -11+140

= 129

This means that for the dollar increase from $55 to $56, Bob's weekly income will increase by an estimated $129 dollars.