Hello, DarkestEvil!

A page contains 32 inē of print.

The margins at the top and the bottom of the page are 2 inches.

The margins on each side are only 1 inch.

Find the dimensions of the page so that the least paper is used.

I was taught to write two equations, one for perimeter and one for area.

No, that is for a specific (and different) problem. Code:

: 1 : - x - : 1 : -
_ *---------------* -
: | | :
2 | | :
: | | :
- | *-------* | :
: | |///////| | :
: | |///////| | :
y | |///////|y | y+4
: | |///////| | :
: | |///////| | :
- | *-------* | :
: | x | :
2 | | :
: | | :
- *---------------* -
: - - x+2 - - :

Let: $\displaystyle x$ = width of printed area.

Let: $\displaystyle y$ = height of printed area.

We are told that: .$\displaystyle xy \,=\,32 \quad\Rightarrow\quad y \,=\, \frac{32}{x}\;\;{\color{blue}[1]}$

The width of the page is: .$\displaystyle x+2$

The height of the page is: .$\displaystyle y+4$

The area of the page is: .$\displaystyle A \:=\:(x+2)(y+4)$

Substitute [1]: .$\displaystyle A \;=\;(x+2)\left(\tfrac{32}{x} + 4\right) $

And we have: .$\displaystyle A \;=\;4x + 64x^{-1} + 40$

And *that* is the function we must minimize.