A book designer has decided that the pages of a book should have margins at the top and bottom and margins on the sides. She further stipulated that each page should have an area of (see the figure). Determine the page dimensions that will result in the maximum printed area on the page.

Question

A book designer has decided that the pages of a book should have  margins at the top and bottom and  margins on the sides. She further stipulated that each page should have an area of  (see the figure).  Determine the page dimensions that will result in the maximum printed area on the page.

Quizplus · Accepted Answer

Let x be the width of the printed area and y be the height of the printed area. 
We know that the total area of the page (including margins) is (x+22)(y+22) since there are 11-inch margins on both sides. 
The printed area is xy. 
So we need to maximize xy subject to the constraint that (x+22)(y+22) = 12100. 
Using the AM-GM inequality: 
(x+22)(y+22) = xy + 22x + 22y + 484 
= xy + 22(x+y) + 484 
≥ 2√(xy * 22(x+y)) + 484 
= 2√(22xy(x+y)) + 484 
So we have: 
2√(22xy(x+y)) + 484 ≤ 12100 
2√(xy(x+y)) ≤ 11616 
√(xy(x+y)) ≤ 5808 
xy(x+y) ≤ 5808^2 
Since we want to maximize xy, we can assume that x = y to simplify the problem. 
Then we have: 
x^2(x+44) = 5808^2 
x^3 + 44x^2 - 5808^2 = 0 
Using a calculator or computer, we can find that x ≈ 104.135. 
So the dimensions of the printed area are approximately 104.135 inches by 104.135 inches. 
Therefore, the correct choice is (C).

A Book Designer Has Decided That the Pages of a Book