Question

What dimensions should you make a rectangular poster of total area 72 square inches in order to maximize the printed area if it is required that the top and bottom margins be 2 inches and the side be 1 inch?

Answer 1

Let's call the width of the poster `x` and the height `y`.

Explanation:

It's fairly easy to see that, because the surface area is
`x*y=72->y=72/x`, but we'll do that later.
Printable in `x`-direction: `x-2*1=x-2`
Printable in `y`-direction: `y-2*2=y-4`
Total printable area: `A=(x-2)*(y-4)=xy-4x-2y+8`

It's now time to substitute the `y`'s:
`A=x*(72/x)-4x-2*(72/x)+8`
`A=(72cancelx)/cancelx-4x-(2*72)/x+8=80-4x-144/x`

For the optimum we need to set the deravative of `A` to `0`

`A=80-4x-144x^-1->A'=0-4-(-1)144x^-2->`
Re-arrange and set to `0`
`144x^-2-4=0or 144/x^2=4->4x^2=144->`
`x^2=144/4=36->x=6` (we only take the positive answer)
This is the width. The height is:
`y=72/x=72/6=12`

Answer : Width is 6", height is 12".

Check :
Total area printed is `(6-2*1)(12-2*2)=4*8=32` sq.in
Graph of the area-function `A=80-4x-144/x`
graph{80-4x-144/x [-36, 112.2, -11.25, 62.75]}