Question

A rectangular box has a square base. The combined length of a side of the square base, and the height is 20 in. Let x be the length of a side of the base of the box.

a. Write a polynomial function in factored form modeling the volume V of the box.

b. What is the maximum possible volume of the box?

c. Explain how you found the maximum possible volume.

Answer 1

Explanation:

V(x) = x3 + 2x2 - 11x - 12

To factor, you want 3 numbers that multiply to give -12 (the constant term) and add to give 2 (coefficient of the x2 term). The answer is 4, -3, and 1: (4)(-3)(1) = -12, 4 + (-3) + 1 = 2. So the volume polynomial factors to:

V(x) = (x+4)(x-3)(x+1)

Now the volume of a rectangular box is:

V = Length*Width*Height = (x+4)(x-3)(x+1)

So let's pick height = (x+1) = 4.5 (you could pick any one of the three factors).

x+1 = 4.5

x = 3.5

Length = (x+4) = 3.5+4 = 7.5

Width = (x-3) = 3.5 - 3 = 0.5

The box's dimensions are 7.5 m by 0.5 m by 4.5 m