You are designing a rectangular poster to contain 64 in2 of

You are designing a rectangular poster to contain 64 in2 of printing with a 4-in margin at the top and bottom and a 1-in margin at each side. What overall dimensions will minimize the amount of paper used? What is the vertical height of the poster that will minimize the amount of paper used?

Solution

printed area = 64 = xy
paper size, P(x)
= (x + 2*1)(y + 2*4)
=(x + 2)(y + 8)
= xy + 2y + 8x + 16
= 8x + 2(64/x) + (64+ 16)
= 8x + 128/x + 80

P\'(x) = 8 - 128/x^2 = 0 ==> 8 x^2 = 128 x^2 = (128/8)
x = (128/8) = (16) =+/- 4  but discard the negative root for a length  (for x > 0)

xy = 64

4y = 64

y = 16

Well, I gave you the minimized x & y , just plug-in (x + 2)(y + 8) ,

answer : paper dimension P = ( 4 + 2 ) ( 16 + 8 ) ==> 6 x 24

P\" = 256 / x^3 (which is positive for positive x, confirming that this will be a min

x is the vertical height of the printed area
4 + 2 = 6 = vertical height of page