The area of the base of an open rectangular box is 120cm2. The front has an area of 96cm2 and the side has an area of 100cm2. What is the volume of the box?
So I was given the surface area of three of the sides of the rectangular prism and I'm trying to figure out how to get the width, height and length only using the surface areas provided to me. I've done some trial and error but no luck. Is there any way I can go about this differently? Thanks!
Area of rectangle is length x height. Volume of rectangular prism is length x width x height.
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$\begingroup$Say the sides of the box are $a, b,$ and $c$.
Then we have $ab=120, ac=96$, and $bc=100$ cm$^2$.
The volume is $abc$ cm$^3$.
Note that $120\times96\times100$ = $ab\times ac\times bc = (abc)^2, $ so
$abc=\sqrt{120\times96\times100}=\sqrt{4\times30\times16\times6\times100}=$
$2\times 4\times 10\times\sqrt{5\times6\times6}=80\times6\sqrt5=480\sqrt5$ cm$^3$.
$\endgroup$ 2 $\begingroup$Let the length, height, width be $L, H, W$ respectively.
Then you have:
$$LW=120\to L=\frac{120}{W}$$$$LH=96\to L=\frac{96}{H}$$$$WH=100$$
Notice from the first two that: $$\frac{120}{W}=\frac{96}{H}\to\frac WH = \frac{120}{96}=\frac 54$$
Use this with the third statement ($WH=100$) to find $W$ and $H$. Then use either of the first two to find $L$, armed with this information.
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