Question

A rectangular cardboard poser is to have 150 sq ft in for printed matter. It is to have a 3inch margin at the top and bottom and a 2inch margin at the right and left. Find the dimensions of the poser so that the amount of cardboard used is minimized.

Answer 1

if the cardboard has height x and width y,

a = xy

printed area is (x-6)(y-4) = 150

so, we want to minimize xy.

a = xy = x(150/(x-6) + 4)
da/dx = 4(x^2-12x-189)/(x-6)^2

since the denominator is never zero, da/dx=0 when
x^2 - 12x - 189 = 0
(x+9)(x-21) = 0

so, x = 21
y = 14

area of poster is 294
area of printing is 15x10 = 150