Find the optimal, width, height, and total area If I was trying to design a printed billboard using a minimal area of plywood. The printed area must be $2000$ sq ft. Here are the margins:
side margins     $10$ ft
top margin   $8$ ft
bottom margin    $6$ ft.
How can I find the optimal, width, height, and total area?
 A: You want to minimize area. Let's call the printed area of dimension $x\times y$ where 
$x$:= width of printed area, 
$y$: = height of printed area.
$xy = 2000\tag{1 printed Area }$
Now, we have to compute the width of the billboard with side margins of 10 feet: $\;w = x + 2\cdot 10 = x + 20$
And the height of the billboard being: $\;y + 8 + 6 = y + 14$
We want to minimize the total area $A$ of the billboard, with $$\;A = (x+ 20)(y+ 14) = 14x + xy + 20y + 280\tag{Billboard Area}$$
Using $(1)$: $y = \dfrac {2000}{x}$
Substituting into the equation for the area of the billboard gives us $$A = 14x + 2000 + 20\cdot \frac{2000}x + 280 = 14x + \dfrac{40000}{x} + 2280\tag{A}$$
Now, find the derivative of $A$, $A'$ with respect to $x$, and set the resulting derivative equal to zero: The solution to that equation will give you the value of $x$ where any minimums and/or maximums are going to occur. You'll need to test each solution (if more than one positive solution exists) to see which is a minumum: which gives you the least possible value when $A$ is evaluated there. 
Then you'll compute the value of $A$, total area, the value of $x + 20$, and the value of $y + 14$.
Feel free to check back once you've differentiated and found the solutions that solve $A' = 0$ (Hint: after derivating, you may want to multiply through by $x^2$ to simplify $A' = 0$, so it is more easily solvable.
