What dimensions should you make a rectangular poster of total area 72 square inches in order to maximize the printed area if it is required that the top and bottom margins be 2 inches and the side be 1 inch?

Question

Aurora Alvarado · Accepted Answer

Let's call the width of the poster #x# and the height #y#.

It's fairly easy to see that, because the surface area is #x*y=72->y=72/x#, but we'll do that later. Printable in #x#-direction: #x-2*1=x-2# Printable in #y#-direction: #y-2*2=y-4# Total printable area: #A=(x-2)*(y-4)=xy-4x-2y+8# It's now time to substitute the #y#'s: #A=x*(72/x)-4x-2*(72/x)+8# #A=(72cancelx)/cancelx-4x-(2*72)/x+8=80-4x-144/x# For the optimum we need to set the deravative of #A# to #0# #A=80-4x-144x^-1->A'=0-4-(-1)144x^-2-># Re-arrange and set to #0# #144x^-2-4=0or 144/x^2=4->4x^2=144-># #x^2=144/4=36->x=6# (we only take the positive answer) This is the width. The height is: #y=72/x=72/6=12# Answer : Width is 6", height is 12". Check : Total area printed is #(6-2*1)(12-2*2)=4*8=32# sq.in Graph of the area-function #A=80-4x-144/x# graph{80-4x-144/x [-36, 112.2, -11.25, 62.75]}

Evelyn Agard · Answer

undefined To maximize the printed area of the rectangular poster given the constraints, you should make the dimensions as follows:

Width: $ w = 72 $ inches
Height: $ h = 36 $ inches