Optimizing Page Dimensions for Efficient Paper Usage | Homework Solution

[xChee]

Homework Statement

The printed area of a page in a book will be 81cm^2. The margins at the top and bottom of the page will each be 3cm deep. The margins at the sides of the page will each be 2cm wide. What page dimensions will minimize the amount of paper?

Homework Equations

Area= length x Width, Perimeter= 2L + 2W

The Attempt at a Solution

I started off by writing a perimeter equation which is, P = 2L+2W. Then came up with a equation for each side length; length was going to be, L+4 and width was going to be, W+6. Then I subbed those equations into P=2L + 2W, which then I got, P= 2(L+4) + 2(W+6). Then I set 81 to the area formula having, 81= LW. After I isolated for w and subbed that into my perimiter equation. After that I found the derivative of it, set it equal to zero and found the dimensions of the paper was 9cm by 9cm... which is wrong...

the right answer is 11.35cm by 17.02cm... I need help D;

here's a picture of the diagram

http://img827.imageshack.us/img827/7226/imageccpl.jpg

 

[Dick]

The perimeter has nothing to do with the problem. Change you variables a little. Call L the total length of the paper and W the total width. Now you know (L-6)(H-4)=81. You want to minimize the total area which is LW.