# Thread: Optimisation - surface area

1. ## Optimisation - surface area

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?

2. Originally Posted by bubbles73
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 $2400=2x^2+4xh$

Solving for h, we get $h=\frac{600}{x}-\tfrac{1}{2}x$

Since $V=x^2h$, we can now substitute the expression we came up with into h:

$V=x^2\left(\frac{600}{x}-\tfrac{1}{2}x\right)=600x-\tfrac{1}{2}x^3$

Differentiate and then set equal to zero.

$V'=600-\tfrac{3}{2}x^2\implies 600=\tfrac{3}{2}x^2\implies 400=x^2\implies x=\pm20$

We take the positive value, since length can't be negative.

Now this implies that $h=\frac{600}{20}-\tfrac{1}{2}(20)=\color{red}\boxed{20}$

Thus, the dimensions of the prism are $\color{red}\boxed{20~cm~\times~20~cm~\times~20~cm}$

This prism is actually a cube.

Does this make sense?

--Chris