List the possible rational solutions of the equation

Question

A pyramid has a square base with sides of length x, a height of x – 2, and a volume of 3. Write a polynomial equation to model the situation. List the possible rational solutions of the equation, and then find the dimensions of the pyramid.

Answer 1

The volume of a right square pyramid is (1/3)x²h where x is the side length of the base. So in this case, since h = x - 2, then V = (1/3)x²(x - 2). Since we are told the volume is 3 then:

3 = (1/3)x²(x - 2)

9 = x²(x - 2)

9 = x³ - 2x²

x³ - 2x² - 9 = 0

To determine the POSSIBLE rational zeros, we can use the rational roots theorem which says that if there are any rational roots, they would be ± the factors of the constant term divided by the leading coefficient so 9 divided by 1, or in other words ±1, ±3, ±9.

Using synthetic division, we find the solution to be x = 3, so the sides of the base are 3 and the height is 1.