6. A solid rectangular brick is to be made from 1 cubic foot of clay. The brick must be 3 times as long as it is wide. For what value of the ratio of height to width will the brick have minimum surface area?

Let,

L = rectangle length; W = rectangle width; H = rectangle height

Where the volume V is given by,

Since, the length is 3 times as long as its width, that is, L = 3W.

We can use equation (1) to solve for H in terms of W as follows,

Substituting, H and L into equation (2) yields,

Thus, to find the value of W that minimizes the surface area, solve the equation A’(W)=0.

Hence, the ratio of height to width is,