Section 4.5

Here are the steps to follow when solving an Applied Maximum/Minimum Problem:
1) Draw a diagram and label the given quantities.
2) Write your “let statements” in sufficient detail so as to completely describe your variables.  Include the proper unit.  You should always have three variables: the two you are being asked to find and the one you are being asked to optimize.
3) Write a formula for the variable you are being asked to optimize.
4) If the formula in step 3 has two variables on the right side of the equation, find a second equation that relates these two variables.  To help find this second equation, ask yourself the question: “What’s the only information in this problem we haven’t used yet?”
5) Solve the second equation for one of its variables, and substitute for this variable in the original formula.  The formula should now express the variable you are trying to optimize as a function of a single independent variable.
6) Determine the interval to which your independent variable (let’s call it x) is restricted.  For the non-omitted homework problems, this should always be a closed interval.  To find the left endpoint, ask yourself the question: “What’s the smallest value x can have?”   To find the right endpoint, ask yourself the question: “What’s the largest value x can have?”
7) At this point, your problem should look identical to those you solved in section 4.4.  Follow the 3-step procedure given in the blue box above example 1 in section 4.4 of your text.
8) Answer the word problem in words, making sure to include the proper units in your answer.

Here are the examples I worked out in class:
1) Find two nonnegative numbers whose sum is 4 such that the sum of their squares is as small as possible.

2) A farmer wants to fence in 60,000 sq. ft. of land in a rectangular plot along a straight highway.  The fence along the highway costs $1 per foot, while the fence for the other three sides costs $0.50 per foot.  How much of each type of fence will have to be bought in order to keep costs to a minimum?

3) A tin can is to be made with a capacity of 2p cubic inches.  What dimensions for it will require the smallest amount of tin?