a)
Let the length of the recatangle be y.
width = x
Thus, the sides of the fence are x, x and y.
Area = length * width = x * y = xy
Perimeter = length + 2(width) = y + 2x
Given that Perimeter = 360, then
y + 2x = 360
y = 360 - 2x
Substituting y = 360 - 2x into area = xy,
Area = (360 - 2x)x
A(x) = 360x - 2x^2
b)
To find the maximum area, we would differentiate A(x) with respect to x. We have
A'(x) = 360 - 4x
For turning point, A'(x) = 0
360 - 4x = 0
4x = 360
x = 360/4
x = 90
Side length x = 90 meters
c) We would differentiate A'(x)
A''(x) = - 4
A''(x) < 0 ; x = 90m is a point of maxima
Maximum area = (360x - 2x^2) at x = 90 m
maximum area = 360(90) - 2(90)^2 = 32400 - 16200
Maximum area = 16200 square meters