The area of the faces of a right rectangular prism are 24, 32, and 48 square centimeters. What is the volume of the prism?

Question

Can someone show me their work, and not just the answer? I need to learn how to do this, and showing work would be greatly appreciated.

Answer 1

Start by assigning variables to things, and writing down equations for what you know.

Let's say the box has length $L$, width $W$, and height $H$. Then, we are given:

$L \cdot W = 24$

$W \cdot H = 32$

$H \cdot L = 48$

We want the volume of the box, which is $V = L \cdot W \cdot H$. Can you figure out how to compute that?

Answer 2

To find the volume given three face areas, we let $a$, $b$, and $c$ be the dimensions of the prism. Now, $ab = 24$, $ac = 32$, and $bc = 48$. Combining them, we get $(abc)^2 = 24 \cdot 32 \cdot 48$. So, $abc = \sqrt{24 \cdot 32 \cdot 48} = \sqrt{36864}$. This is exactly 192.

To solve for any volume given 3 faces, where $x$, $y$ and $z$ are face areas, the formula is $V = \sqrt{xyz}$.