Question

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?

Answer 1

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`