A rectangular page is to contain 45 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Answer 1

sqr45 = 3sqr5 = about 3(2.2) = 6.6 for sides of the printed area

8.6 x 8.6 inches for the full page dimensions, rounded off to nearest tenth of an inch

or exactly 2+3sqr5 by 2+3sqr5

a square minimizes the perimeter for a given area

Area = 45 = LW

L=45/W

(L+2)(W+2) = area of the full page

=(45/W+2)(W+2) = 45+2W +90/W + 4

Area = full page = 2W + 49 + 90/W

take the derivative and set it equal to zero and solve for L

A' = 2 -90/W^2 = 0

90/W^2 = 2

W^2 =45

W = sqr45 = 3sqr5 = about 6.6

L= 45/sqr45 = sqr45 = about 6.6

W+2 = 8.6

L+2 = 8.6

dimensions of full page are 8.6 x 8.6