Answer:
1) x = 50 m
2) Max area = 22500 m²
3) Max area of each pen = 1500 m²
4) y = 150 m
5) dimensions of pen are 50m and 30m
6) Graph is attached
Explanation:
Area of a rectangle is length × width
Let length be y and let width be x
Thus;
A = xy
Fencing material is 600m
Thus,perimeter = 600
Perimeter is; P = 6(length) + 2(width)
Thus;
600 = 6x + 2y
y = (600 - 6x)/2
y = 300 - 3x
Thus,
A = x(300 - 3x)
A = -3x² + 300x
We can see that the coefficient of x² is negative. This means that the parabola will open downward and has a maximum value at the vertex.
The vertex formula is;
x = -b/2a
a = -3
b = 300
Thus;
x = -300/(2 × -3)
x = 50 m
Since y = 300 - 3x, then;
y = 300 - 3(50)
y = 150 m
Maximum area of entire garden = xy = 50 × 150 = 7500 m²
There are 5 pens, thus;
Max area of each pen = max area of entire garden/5 = 7500/5
Max area of each pen = 1500 m²
Dimensions of each pen will be;
x and y/5
>> x = 50 m and y/5 = 150/5 = 30m