A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

l = 4 in d = 6 in Note : l represent the width and d represent the height

Step-by-step explanation:

Lets asume the dimensions of a rectangular page is l * d where l is the wide and d is top-bottom leght

So total area of the page is A = l*d ⇒ d = A/l ⇒ d = 8/l

Then the print area of the page will be

l = x + 2 (wide) d = y + 4 (vertical lenght)

So area of the page is

A (x) = (x + 2) * (y + 4) but y = 8/l and l = x + 2 ⇒ y = 8 : (x + 2)

A (x) = (x + 2) * (8 / x + 4) ⇒ A (x) = 8 + 4x + 16/x + 8 ⇒A (x) = 4x + 16 / x

Taken derivative we have:

A' (x) = 4 + (-1 * (16) / x² ⇒ A' (x) = 4 - 16/x²

A ' (x) = 0 means 4 - 16 / x² = 0 ⇒ 4 x² - 16 = 0 x² = 4 x = 2

Therefore y = 8 : (x + 2) and y = 2

And the dimensions of the page is

l = 4 in d = 6 in