1. Polynomials: Application
Example: A box with a square base and no top is to be built with a capacity of 360 cubic inches. If the material for the sides costs 2 cents per square inch and the material for the bottom costs 3 cents per square inch, find the dimensions that will result in a total cost of $5.88 to make the box.
Let x represent the width (and length) of the box and y the height.
The volume, V, of the box is (length)(width)(height),
Since the volume of the box is 360 cubic inches, 360 = x2y.
so V = (x)(x)(y), V = x2y.

2. Polynomials: Application
The cost, C, (in dollars) to make the box is (cost per sq inch of the bottom)(# of sq inches of bottom) + 4(cost per sq inch of a side)(# of sq inches of a side).
Therefore C = 0.03x (0.02)(xy).
Substituting here results in:
However, this is a cost function of two variables (x and y).
To turn it into a cost function of one variable (x) use the volume equation, 360 = x2y.
From this,
Slide 2

3. Polynomials:Application
Simplifying the second term results in:
Substituting the total cost of $5.88 to make box for C
results in:
Multiplying each term by x results in needing to find the zeros of a third degree polynomial.
5.88x = 0.03x ,
0 = 0.03x3 – 5.88x
Slide 3

4. Polynomials:Application
0 = 0.03x3 – 5.88x
Since only approximate solutions are needed, this equation can best be solved graphically.
The solutions are x = 6 inches and x = 10 inches.
To find y, substitute into
So the box either has dimensions of 6 by 6 by10 inches
or dimensions of 10 by 10 by 3.6 inches.
Slide 4

5. Polynomials:Application
Try: A rectangular box with no top will be made from a sheet of notebook paper (size 8½ by 11 inches) by cutting out squares (each side of length x) from each corner and folding up the sides as shown. x
Find the value of x that produces a box with a volume of cubic inches. x
Answer: x = 1.3 inches.
Slide 5

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Polynomials:Application
END OF PRESENTATION
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