Question

A rectangular page is to contain 72 square inches of print. The page has to have a 4 inch margin on top and at the bottom and a 2 inch margin on each side. Find the dimensions of the page that minimize the amount of paper used.

Answer 1

let the piece of paper be x inches wide and y inches long
area of printed part = (x-4)(y-8)
= xy - 8x - 4y + 32
but that equals 72
xy - 8x - 4y + 32 = 72
y(x-4) = 8x + 40
y = (8x+40)/(x-4) , clearly x > 4

amount of paper used = A
A = xy
= x(8x+40)/x-4)
= (8x^2 + 40x)/(x-4)

dA/dx = ((x-4)(16x+40) - (8x^2 + 40x)(1) )/(x-4)^2
= 0 for a min of A

(16x^2 - 24x - 160 - 8x^2 - 40x)/(x-4)^2 = 0
8x^2 - 64x - 160 = 0
x^2 - 8x - 20 = 0
(x - 10)(x + 2) = 0
x = 10 or x = -2 , the latter is not possible

x = 10 , then y = (8(10) + 40)/(10-4) = 20

The sheet of paper should be 10 inches wide and 20 inches long

Answer 2

A page is to contain 24 sq.in. of print. The margins at top anc bottom are 1.5 in. At the side 1 in. Find the most econonmical for the page