Rectangular box with a mean-proportional edge

Text

A rectangular box has two edges $12$ cm and $27$ cm; the third edge is the mean proportional of the two. Find the total surface area and the volume.

Solution

The third edge is $c=\sqrt{12\cdot27}=\sqrt{324}=18$ cm. Volume: $V=12\cdot27\cdot18=5832\ \text{cm}^3.$ Total surface area: $S=2(12\cdot27+12\cdot18+27\cdot18)=2(324+216+486)=2052\ \text{cm}^2.$ $\boxed{S=2052\ \text{cm}^2;\ \ V=5832\ \text{cm}^3}$