A square-based rectangular prism has a total surface area of 2400cm^2. What are the dimensions of the prism if its volume is a maximum?
Let x represent the sides of the base of the square-based rectangular prism, and let h be the height of the prism.
We are told that the surface area is 2400 cm^2.
The equation we will use to define surface area will be $\displaystyle 2400=2x^2+4xh$
Solving for h, we get $\displaystyle h=\frac{600}{x}-\tfrac{1}{2}x$
Since $\displaystyle V=x^2h$, we can now substitute the expression we came up with into h:
$\displaystyle V=x^2\left(\frac{600}{x}-\tfrac{1}{2}x\right)=600x-\tfrac{1}{2}x^3$
Differentiate and then set equal to zero.
$\displaystyle V'=600-\tfrac{3}{2}x^2\implies 600=\tfrac{3}{2}x^2\implies 400=x^2\implies x=\pm20$
We take the positive value, since length can't be negative.
Now this implies that $\displaystyle h=\frac{600}{20}-\tfrac{1}{2}(20)=\color{red}\boxed{20}$
Thus, the dimensions of the prism are $\displaystyle \color{red}\boxed{20~cm~\times~20~cm~\times~20~cm}$
This prism is actually a cube.
Does this make sense?
--Chris