I need help with this question

Question

A rectangular page is designed to contain 121 sqr in. Of print. The margins at the top and bottom of the page are each 1 in deep. The margins of each side are 1.5 in wide. What should the dimensions of the page be so that the least amount of paper is used?

Answer 1

Given:

Rectangular page contains 121 in^2 of print.

Margins at top and bottom are 1 inch

Side margins are 1.5 inch

Find: Dimensions of page to minimize amount of paper.

Solution:

We want to minimize the size of the paper.

So, develop an equation for the size of the paper.

Then, graph the function and find the minimum.

y = size of paper

y = (height)*(width)

y = (2 + a)*(3 + b)

Where a*b = area of printed area.

a*b = 121

If a = x, then x*b = 121. And b = 121/x

Therefore:

a*b = (x)*(121/x)

y = (2 + a)*(3 + b)

y = (2 + x)*(3 + 121/x)

y = 6 + 242/x + 3x + 121

y = 3x + 242/x + 127

Use a graphing calculator to graph it.

You will find a minimum at x = 8.98

So, a = x = 8.981

and b = 121/x = 121/8.981 ~ 13.47

Recall:

y = size of paper

y = (height)*(width)

y = (2 + a)*(3 + b)

So, the paper dimensions are:

Height = 2 + 8.98 ~ 10,98 in

Width = 3 + 13.47 ~ 16.47 in

Total Area = (10.98)*(16.47) = 180.84 in^2

NOTE: If we let a = 121/x and b = x

Then:

y = (2 + 121/x)*(3 + x)

y = 127 + 2x + 363/x

Minimum at x = 13.47

Then:

Height = 2 + 13.47 = 15.47

Width = 3 + 121/13.47 = 11.98

Total Area = (15.47)*(11.98) = 186.53 in^2 WORSE!!

************** Therefore: ****************

Height = 10.98" and Width = 16.47"

Total Area = 180.84 in^2

******************************************