A rectangular page is to contain 24 square inches of print. The margin at the top and at the bottom of the page are 1.5 inches, and the margins on the left and on the right are 1 inch wide. What should be the dimensions of the paper so that the least amount of paper is used.

Dimensions of the paper

height h = 9 in

length L = 6 in

A (min) = 54 in²

Step-by-step explanation:

Let x and y be dimensions of the print area then

x*y = 24 in² then y = 24/x

The total area is:

Heigth (h) h = y + 2*1.5 h = y + 3

lenght (L) L = x + 2

A (page) = h*L then

A = (y + 3) * (x + 2)

A (x) = (24/x + 3) * (x + 2)

A (x) = 24 + 48/x + 3x + 6

A (x) = 30 + 48 / x + 3x

Taking derivatives on both sides of the equation

A' (x) = - 48/x² + 3

A' (x) = 0 - 48/x² + 3 = 0 - 48/x² = - 3

x² = 48/3 x² = 16 x = 4 in

and y = 24 / 4 = 6 in

Then dimensions of paper

h = y + 3

h = 9 in

L = x + 2

L = 4 + 2

L = 6 in

A (min) = 9*6

A (min) = 54 in²