Suppose you have 200 feet of fencing to enclose a rectangular plot. How do you determine dimensions of the plot to enclose the maximum area possible?

1 Answer
May 3, 2017

Answer:

The length and width should each be #50# feet for maximum area.

Explanation:

The maximum area for a rectangular figure (with a fixed perimeter) is achieved when the figure is a square. This implies that each of the 4 sides are the same length and #(200" feet")/4=50" feet"#
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Suppose we didn't know or didn't remember this fact:

If we let the length be #a#
and the width be #b#
then
#color(white)("XXX")2a+2b=200# (feet)

#color(white)("XXX")rarr a+b=100#
or
#color(white)("XXX")b=100-a#

Let #f(a)# be a function for the area of the plot for a length of #a#
then
#color(white)("XXX")f(a)=axxb=axx(100-a)=100a-a^2#

This is a simple quadratic with a maximum value at the point where it's derivative is equal to #0#

#color(white)("XXX")f'(a)=100-2a#

and, therefore, at it maximum value,
#color(white)("XXX")100-2a=0#

#color(white)("XXX")rarr a=50#

and, since #b=100-a#
#color(white)("XXX")rarr b=50#