Question

A fence is to be built to enclose a rectangular area of 1800 square feet. The fence along three sides is to be made of material that costs ​$6 per foot. The material for the fourth side costs ​$18 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built.

Answer 1

Answer:

length = 60 foot, width = 30 foot

Step-by-step explanation:

Area of rectangular part, A = 1800 ft²

Cost of fencing three sides is $ 6 per foot and cost of one side fencing is $18 per foot.

Let the length of the rectangle is l and the width of the rectangle is W.

Area = Length x width

A = L x W

1800 = L x W ...... (1)

Total cost of fencing, C = 6 x ( L + W + L) + 18 x W

C = 6 (2L + W) + 18 W

C = 12 L + 24 W

Substitute the value of W from equation (1), $W=\frac{1800}{L}$ in equation (2)

$C=12L+24\times \frac{1800}{L}$

$C=12L+\frac{43200}{L}$

Differentiate both sides with respect to L:

$\frac{dC}{dL}=12-\frac{43200}{L^{2}}$

Put it equal to zero for maxima and minima

$12-\frac{43200}{L^{2}}=0$

L = 60 foot

and W = 30 foot

So, the costing is minimum for length = 60 foot and the width = 30 foot.