Maria has several containers that are in the shape of a rectangular prism with a square base. The containers all have dimensions such that the edges of the square base are centimeters less than the container’s height, as shown in the figure. Which of the following functions could model the volume of a container, in cubic centimeters, where represents the height, in centimeters, of the container?

Question

Maria has several containers that are in the shape of a rectangular prism with a square base. The containers all have dimensions such that the edges of the square base are centimeters less than the container’s height, as shown in the figure. Which of the following functions could model the volume of a container, in cubic centimeters, where represents the height, in centimeters, of the container?

Answer 1

Let's call the length of one side of the square base x (in centimeters) and the height of the container h (in centimeters).

Then, the volume of the container can be represented by the formula:

V = x^2 * h

Given that the edges of the square base are x - 3 centimeters less than the height (h), we can write:

x = h - 3

Substitute x = h - 3 into the volume formula:

V = (h - 3)^2 * h
V = (h^2 - 6h + 9) * h
V = h^3 - 6h^2 + 9h

Therefore, the function that models the volume of the container in terms of h is V = h^3 - 6h^2 + 9h.