Question:

A rectangular box with an open top is to be constructed from a rectangular sheet of cardboard that measures 24 inches by 36 inches. The box has a square base with side length $x$ and a height of $h$.

(a) Express the volume $V$ of the box as a function of $x$.

(b) Express the surface area $A$ of the box as a function of $x$.

(c) Determine the dimensions for the box that maximize the volume. What is the maximum volume?

Answer:

(a) To express the volume $V$ of the box as a function of $x$, we need to find the area of the base and multiply it by the height.

The area of the base is $x^2$ and the height is $h$. Therefore, the volume $V$ is given by:

The area of the base is $x^2$. The four lateral faces each have dimensions $x$ by $h$. The top face has dimensions $x$ by $36$ (the longer side of the cardboard sheet). Therefore, the surface area $A$ is given by:

Since the box has a square base, we have $x = h$. The length of the cardboard sheet is used for the base and the height combined, so we also have $2x + h = 36$. Substituting $x$ for $h$, we get:

Therefore, the dimensions that maximize the volume of the box are $x = h = 12$.

To find the maximum volume, we substitute $x = 12$ into the volume function:

Since $h = x$, we have $V = 144h$.

So, the maximum volume is $V = 144(12) = 1728$ cubic inches.

To summarize:

(a) The volume $V$ of the box as a function of $x$ is given by: