Answer:
The length of the rectangular pen is [tex]40\ ft[/tex]
The width of the rectangular pen is [tex]20\ ft[/tex]
Step-by-step 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
[tex]P=x+2y[/tex] ---> remember that one side of the rectangle won’t need a fence
[tex]P=80\ ft[/tex]
so
[tex]80=x+2y[/tex]
[tex]y=(80-x)/2[/tex] -----> equation A
The area of the rectangular pen is equal to
[tex]A=xy[/tex] -----> equation B
Substitute equation A in equation B
[tex]A=x*(80-x)/2\\ \\A=-0.5x^{2}+40x[/tex]
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
[tex]x=40\ ft[/tex]
Find the value of y
[tex]y=(80-x)/2[/tex] ----> [tex]y=(80-40)/2=20\ ft[/tex]
therefore
The length of the rectangular pen is [tex]40\ ft[/tex]
The width of the rectangular pen is [tex]20\ ft[/tex]
