Box with proportional dimensions and given edge sum

Text

In a rectangular box the sum of all $12$ edges is $96$ cm and the dimensions are proportional to $3$, $4$, $5$. Find the total surface area and the volume.

Solution

The edge sum is $4(a+b+c)=96\Rightarrow a+b+c=24$. With ratios $3:4:5$ (total $12$): $a=6,\quad b=8,\quad c=10.$ Total surface area: $S=2(ab+ac+bc)=2(48+60+80)=376\ \text{cm}^2.$ Volume: $V=6\cdot8\cdot10=480\ \text{cm}^3.$ $\boxed{S=376\ \text{cm}^2;\ \ V=480\ \text{cm}^3}$