Min-Max ProblemsUnit 3 Lesson 2b – Optimization Problems

Finding Solutions to Min-Max Problems

Min-Max Problems – How to solve these problems:

1) Read the problem carefully. Distinguish constants from variables.

2) Decide what is being maximized or minimized. This gets the

dependent variable.

3) Write the quantity as a function of 1 variable. Use other information

in the problems and your knowledge of formulas to help you do this.

Min-Max Problems – How to solve these problems (cont.):

4) Determine an appropriate domain if necessary.

5) Graph your equation. Determine maxima and / or minima. You may

need to adjust your viewing window on your calculator to see this.

Keep the domain in mind as well.

6) Make sure you answer the question that was asked.

Finding Solutions to Min-Max Problems

Example 1: Maximize Volume

• A box manufacturer is asked to make a box from a rectangular piece of corrugated cardboard in the length of 20 inches by 30 inches. What size should the height (x) of the box be to maximize the volume of the box?

20”

30”

x

x

Example 2: Minimize Surface Area

• A container manufacturer wants a cylindrical metal can that will hold one quart of paint with a little extra room for air space. One liquid quart occupies a volume of 57.749 in3, so the manufacturer decides to design a can with a volume of 58 in3. The manufacturer wants to keep the cost of material at a minimum. The amount of sheet metal required to construct the can is equal to the total surface area A of the can. Let r be the radius and h be the height of the can. The problem is to find a pair of values for the radius and height which give a volume of 58 in3 and a minimum value for A, the surface area.

Closure – Solving Min-Max Problems

• When solving min-max problems what are some steps you need to take?• How can you find the correct window to view your graph?• Why is knowing the domain important?