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?