Determine the page dimensions that will minimize the amount of paper used?A designer is designing a layout on a rectangular page that will contain...

The area of the paper is of 24 square
inches.

A = x*y

24 = x*y
=> y = 24/x

The dimensions of the rectangular layout
are:

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

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

To determine the dimensions that will
minimize the amount of paper, we'll have to calcualte the 1st derivative of the area
function.

We'll use the product
rule:

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

A'(x) = 24/x + 3-
24(x+2)/x^2

A'(x) = (24x + 3x^2 - 24x -
48)/x^2

We'll eliminate like
terms:

A'(x) = ( 3x^2 -
48)/x^2

We'll cancel
A'(x);

A'(x) = 0

( 3x^2 -
48)/x^2 = 0 => 3x^2 - 48 = 0 => 3x^2 = 48 => x^2 = 16 => x1
= 4, x2 = -4

Since a dimension cannot be negative, we'll
reject the negative value -4.

y = 24/4 => y =
6

Therefore, the minim dimensions of the
paper are: x = 4 and y = 6.