A farmer wants to build a rectangular pen with 80 feet of fencing. The pen will be built against the wall of the barn so one side of the rectangle won’t need a fence. What dimen…

Question

asked Feb 28, 2020 196k views

2 Answers

Musa Usman · Answer 1 · 2020-03-06T14:21:12+0000

Answer:

The length of the rectangular pen is
$40\ ft$

The width of the rectangular pen is
$20\ ft$

Explanation:

Let

x-----> the length of the rectangular pen

y----> the width of the rectangular pen

we know that

The perimeter of the rectangular pen in this problem is equal to

P=x+2y ---> remember that one side of the rectangle won’t need a fence

$P=80\ ft$

so

80=x+2y

y=(80-x)/2 -----> equation A

The area of the rectangular pen is equal to

A=xy -----> equation B

Substitute equation A in equation B

$A=x*(80-x)/2\\ \\A=-0.5x^(2)+40x$

The quadratic equation is a vertical parabola open downward

The vertex is a maximum

The x-coordinate of the vertex is the length of the rectangular pen that maximize the area of the pen

The y-coordinate of the vertex is the maximum area of the pen

using a graphing tool

The vertex is the point (40,800)

see the attached figure

so

$x=40\ ft$

Find the value of y

y=(80-x)/2 ---->
$y=(80-40)/2=20\ ft$

therefore

The length of the rectangular pen is
$40\ ft$

The width of the rectangular pen is
$20\ ft$