# Problem of the Week Problem D and Solution I Want Some Volume

## Problem

The areas of the front, side, and top faces of a rectangular prism are $$2xy$$, $$\dfrac{y}{3}$$, and $$96x\text{ cm}^2$$, respectively.

Calculate the volume of the rectangular prism in terms of $$x$$ and $$y$$.

## Solution

Since $$\frac{y}{3}$$ and $$96x$$ are areas, then $$x$$ and $$y$$ must be positive. Let the length, width, and height of the rectangular prism be $$a$$, $$b$$, and $$c$$, respectively.

The volume is equal to the product $$abc$$.

By multiplying side lengths, we can write the following three equations using the given areas. \begin{aligned} ac &= 2xy\\ bc &= \frac{y}{3}\\ ab &= 96x\end{aligned} Multiplying the left sides and multiplying the right sides of each of the three equations gives us the following. \begin{aligned} (ac)(bc)(ab) &= (2xy)\left(\frac{y}{3}\right)(96x)\\ a^2b^2c^2 &= 64x^2y^2\\ (abc)^2 &= (8xy)^2\\ \sqrt{(abc)^2} &= \pm~\sqrt{(8xy)^2}\\ abc &= \pm ~8xy\end{aligned} Since all quantities are positive, we can conclude that $$abc=8xy$$.

Therefore, the volume of the rectangular prism is $$8xy~\text{cm}^3$$.