3 7 Optimization Problems Finding Maximum Volume A

3. 7 Optimization Problems

Finding Maximum Volume • A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume?

• Because the box has a square base, its volume is • Surface area of the box is

Because we are trying to maximize volume…. .

• Since x cannot be negative, we use 6. • You can check it by taking the second derivative test.

Finding Minimum Distance • Which points on the graph of are closest to the point ?

Finding Minimum Distance Using the equation, • Because d is the smallest when the expression inside the radical is smallest, you need to find the critical numbers of

Finding Minimum Area • A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1. 5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

Let A be the area to be minimized

Finding Mimimum Length • Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?

Let W be the wire length to be minimized.

So, W is

An Endpoint Maximum Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the circle to enclose the maximum total area?

A=(area of square) + (area of circle) Because the total length of wire is 4 feet, So, then……

An Endpoint Maximum continued The only critical number in (0, 1) is So, We can conclude that the maximum area occurs when x=0. That is, all the wire is used for the circle.