3.
9 Optimization
Optimization is one of the most common applications in calculus. We often hear terms like least
time, greatest profit, shortest distance, maximum volume, minimum waste. In order to optimize
problems of this sort, we first need to find the quantity that needs to be minimized or maximized.
Be sure to assign variables and write a function that represents the minimum or maximum needed.
Then, use the techniques we’ve learned in this chapter.
Primary equation: An equation that gives a formula to the quantity to be optimized.
Secondary equation : An equation that provides information needed to complete the primary
equation. This equation will determine domain constraints.
EX. #1: Find two positive numbers such that the sum of the first and twice the second is 64 and
whose product is a maximum.
EX #2: A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain
180,000 square meters in order to provide enough grass for the herd. What dimensions would
require the least amount of fencing if no fencing is needed along the river?
© 2012 - 2016 Flamingo MathTM (Jean Adams)
EX #3: An open box is being constructed from a piece of sheet metal 18 inches by 30 inches by cutting
out squares of equal size from the corners and bending up the sides. What size squares should
be cut to make a box of maximum volume? What is the volume?
EX. #4: What are the dimensions of an aluminum can that can hold 40 𝑖𝑖𝑛𝑛3 of soda and that uses
the least amount of aluminum? Assume the can is cylindrical and capped on both ends.
© 2012 - 2016 Flamingo MathTM (Jean Adams)
EX. #5: Which points on the graph of 𝑦𝑦 = 4 − 𝑥𝑥 2 are closest to the point (0, 2)?
© 2012 - 2016 Flamingo MathTM (Jean Adams)
