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?

1 Answer
Nov 4, 2015

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

Explanation:

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]}