Download Handout11-6b

Chapter 11 – 6b Related Rates
Example 1. For a company manufacturing calculators, the cost and revenue functions are given by:
𝐶 = 90,000 + 30𝑥 and 𝑅 = 300𝑥 −
𝑥2
30
where x is the number of calculators made in one week. Suppose the company CEO decides to increase
production from 1,500 calculators per week to 1,600 calculators per week. Use the instantaneous rate of
change in the cost and revenue functions with respect to time to estimate the cost, revenue, and profit
after the change and compare your results to the actual values.
Example 2. A retail store estimates that weekly sales s and weekly advertising costs x (both in dollars)
are related by:
𝑠 = 60,000 − 40,000𝑒 −0.0005𝑥
Suppose the current advertising costs are $2,000 per week. Use implicit differentiation to estimate the
rate of increase in sales per week if the advertising costs are raised to $2,300 a week. Compare this
estimate to the result you get using the equation above.
Example 3. The price p and the demand x for a product are related by:
2𝑥 2 + 5𝑥𝑝 + 50𝑝2 = 80,000
A) If the price is $30 per unit and increasing at the rate of $2 per month, what is the instantaneous rate
of change in the demand per month?
B) If the demand is 150 units per month and decreasing at the rate of 6 units per month, what is the rate
of change in the price per month?
Example 4. An oil tanker has run aground on a reef and is leaking which has resulted in a circular pool of
oil about an inch thick around the ship. If the radius of the oil spill is 500 feet and increasing at the rate
of 4 inches per minute, how fast is oil leaking from the tanker (in cubic feet per minute)?