## 3.6 Optimization

### Summary

Another application of mathematical modeling with calculus involves word problems that seek the largest or smallest value of a function on an interval. This class of problems is called optimization problems; an introduction and space for practice are provided in this section.

By the end of your studying, you should know:

• How to set up and solve optimization word problems.

On-screen applet instructions: Shown is a rectangle of fixed perimeter. Use the slider to find experimentally the length and width that maximize the area.

### Examples

You want to run an underground power cable from a power station on one side of a river to a house on the other side. The house is 5 miles downstream from the station, and the river has a constant width of 1 mile. It costs \$1000 per mile to lay cable underground, and \$3000 per mile to lay cable under water. How should you lay the cable to minimize the total cost, and what will the minimum cost be?

You want to smuggle a precious metal out of the country, by disguising it as a single cylindrical barrel, closed at both ends. The cost of shipping is \$7 per cubic foot. Once out the the country, you can sell the metal for \$8 per square foot. Assuming that you design the barrels with the height equal to twice the diameter, how many square feet should you smuggle, and what will your profit be?

A wire 50 inches long is cut into two pieces. One piece is bent into a circle; the other, into a square. Where should the wire be cut to minimize the sum of the areas of the two shapes?

Optimization

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 3.6 Optimization (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

### Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Optimization
Kouba

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel