Question

A rectangular page is to contain 24 square inches of printed material. The margins at the top and at the bottom are each 1.5 inches. The margins at the left and at the right are each 1 inch. What dimensions should the page have so that the least amount of material is used?

Answer 1

Height: 9 in

Length: 6 in

Width: 4 in

Explanation:

Given that A rectangular page is to contain 24 square inches

• Let y is the length of the rectangle
• Let x is the width of the rectangle

We know that:

The area of the the rectangle: A = length* width =24

<=> x*y = 24

<=> y = 24/x

As given in the question:

• Height (h) h = y + 2*1.5 <=> h = y + 3
• Lenght (l) = x + 2

<=> The area = h*l

= (y + 3 ) (x + 2 )

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

= 30 + 48 /x + 3x

Taking derivatives on both sides of the equation

A´(x) = -48/x² + 3

Let A´(x) = 0, we have:

-48/x² + 3 = 0

<=> x = 4 in

When x=4 in we have y = 24/4 = 6 and h = 6+3 = 9

=> A (min) = 6*9 = 54

So the dimensions are:

Height: 9 in

Length: 6 in

Width: 4 in