Let
x--------> the length side of the rectangle
y-------> the width side of the rectangle
see the attached figure N 1 to better understand the problem
we know that
the perimeter of the rectangle is equal to
[tex]P=2x+5y\\ P=30,000\ ft[/tex]
so
[tex]2x+5y=30,000\\ 5y=30,000-2x[/tex]
[tex]y=6,000-0.40x[/tex] --------> equation 1
the area of the rectangle is equal to
[tex]A=x*y[/tex] -------> equation 2
substitute equation 1 in equation 2
[tex]A=x*[6,000-0.40x][/tex]
[tex]A=-0.40x^{2} +6,000x[/tex]
Using a graph tool
see the attached figure
The vertex of the function is the point with the maximum area
the vertex of the function is the point (7,500, 22,500,000)
that means that
for x=7,500 ft
the maximum area is 22,500,000 ft^2
Find the value of y
[tex]y=6,000-0.40*7,500[/tex]
[tex]y=3,000\ ft[/tex]
the dimensions of the rectangle are 7,500 ft * 3,000 ft
the maximum area of each pasture is
(22,500,000 ft^2)/4=5,625,000 ft^2
therefore
the answer is
the maximum area of each pasture is 5,625,000 ft^2
