Page Poster

Posted in Uncategorized on 05/16/2010 03:11 pm by admin

Page Poster
what are the dimensions of the page which will minimize the amount of paper used?

A rectangular poster is to contain 144 square inches of print. The margins at the top and bottom and on each side are to be 6 inches. Find the dimensions of the page which will minimize the amount of paper used.

let the dimensions of the poster be x by y, then the total paper used is xy
now the print area is given by (x-12) * (y-12) thus we have
(x-12) * (y-12) = 144
y-12 = 144/(x-12)
y = 12 + 144/(x-12)
thus we want to minimize
A(x) = 12x + 144x/(x-12)
to do this we solve
A’(x) = 0
A’(x) = 12 + 144/(x-12) – 144x/(x-12)^2 = 0
12(x-12)^2 + 144(x-12) – 144x = 0
12x^2 – 288x = 0
12x^2 = 288x
x = 24
and thus
y = 12 + 144/12 = 12 + 12 = 24
thus the optimal dimensions are 24 by 24 giving a total area of 576

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