Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is $84$
A) $7$ , $84$ , $7$
B) 4, 8, 16
C) $7$ , $7$ , $7$
D) 32, $7$ , 16
E) 32, 32, 32

Question

Quizplus · Accepted Answer

Let the dimensions of the rectangular box be $x$, $y$, and $z$. Then we have $4x+4y+4z=84$, or $x+y+z=21$. We want to maximize the volume of the box, which is given by $xyz$. From $x+y+z=21$, we can use AM-GM to get $$\frac{x+y+z}{3}\geq \sqrt[3]{xyz}.$$ Therefore, $xyz\leq \frac{1}{27}(x+y+z)^3$. Since we want to maximize $xyz$, we want $x$, $y$, and $z$ to be as close together as possible. The only way to do this while keeping $x+y+z=21$ is to have $x=y=z=7$. Therefore, the dimensions are $7,7,7$, which is choice $\boxed{\textbf{(C)}}$.

Find the Dimensions of a Rectangular Box of Maximum Volume 848484

Find the Dimensions of a Rectangular Box of Maximum Volume $84$