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.

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 = 0.03x2 + 4(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

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 = 0.03x3 + 28.8, 0 = 0.03x3 – 5.88x + 28.8 Slide 3

Polynomials:Application 0 = 0.03x3 – 5.88x + 28.8 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

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 64.428 cubic inches. x Answer: x = 1.3 inches. Slide 5

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