You will be solving optimization problems.

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Draw a diagram, figure, and/or graph for each problem and label it accordingly.

Show all the mathematical steps involved.

Verify your answer using the second derivative.

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1. An open rectangular box is to be made from a piece of cardboard 8 in. wide and

15 in. long by cutting a square from each corner and bending up the sides.

Find the dimensions of the box of largest volume.

(a) What are the appropriate labeled diagrams (draw them)?

FIGURE 1 FIGURE 2

The flat piece of cardboard with The sides folded up and taped

the corners marked for cutting to make the box with no top

(b) What is to be maximized?

(c) What is the equation for the answer to (b)?

(d) What is the derivative of the equation in (c)?

(e) If we use x as the length of the side of one of the square corners that are cut out,

what is the maximum value x can be? In other words, x < ____?

(f) What is the derivative of the equation in (d)? (which is the second derivative of the

equation in (c))

(g) For what value of x is the second derivative less than 0? In other words, x < ____?

(This tells us what value x must be to create the maximum volume.)

(h) What are the values of x when we solve for the first derivative equal to 0?

(i) Based on the answers to (e), (g), and (h), what is x?

(j) The dimensions are:

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2. One side of an open field is bounded by a straight river.

How would you put a fence around the other three sides of a rectangular plot in

order to enclose as great an area as possible with a given length of fence?

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3. An open storage bin with square base and vertical sides is to be constructed from

a given amount of material. (Neglect the thickness of the material and waste in

construction.)

Determine the dimensions if its volume is to be a maximum.