Answer:
The dimensions of the garden that will minimize the costs are: x=10ft, y=20ft
Step-by-step explanation:
Let x be the length of the brick wall. Let y be the length of the adjacent sides.
Area of the garden = 200 ft
Therefore: xy=200
[TeX]y=\frac{200}{x}[/TeX]
Perimeter of the Fence = Length of
brick wall+Length of Opposite Metal Fence + Length of Other Adjacent Metal Fence
=x + x + 2y
Costing of brick wall= $12 per foot Cost of Metal =$4 per foot.
Cost of the Fence=12x+4x+4(2y)
=16x+8y
Recall: [TeX]y=\frac{200}{x}[/TeX]
Therefore, the cost in terms of x is:
[TeX]C(x)=16x+8(\frac{200}{x})[/TeX]
[TeX]C(x)=\frac{16x^2+1600}{x}[/TeX]
The minimum costs C(x) occur when its derivative equals zero.
[TeX]C^{'}(x)=\frac{16x^2-1600}{x^2}[/TeX]
Thus:
[TeX]16x^2-1600=0[/TeX]
[TeX]16x^2=1600[/TeX]
[TeX]x^2=100[/TeX]
x=10ft
Recall that:
[TeX]y=\frac{200}{x}[/TeX]
y=200/10=20 ft
The dimensions of the garden that will minimize the costs are: x=10ft, y=20ft
