Question

A rectangular field with one side along a river is to be fenced. Suppose that no fence is needed along the river, the fence on the side opposite the river costs $40 per foot, and the fence on the other sides costs $10 per foot. If the field must contain 57,800 square feet, what dimensions will minimize costs?

Answer 1

Answer:

to minimize the cost the dimensions should be 340 feet on sides and 170 on the opposite side of the bank

Step-by-step explanation:

Let the length of rectangular field be 'L'

and width be 'B'

Therefore, the fence required = 2B + L

Total cost of fence, C = $40L + $10 × 2B = $40L + $20B           ............(1)

Area of the field = 57,800 square feet

also, area of the field = B × L = 57,800

or

L = $\frac{57,800}{B}$

substituting L in (1)

C =  $\ 40\frac{57,800}{B}+\ 20B$

for minimizing the cost

$\frac{dC}{dB}$=$\frac{d(\ 40\frac{57,800}{B}+\ 20B)}{dB}$=0

or

0=$-\ 40\frac{57,800}{B^2}+\ 20$

or

$\ 40\frac{57,800}{B^2}=\ 20$

or

$\ 40\times57,800=\ 20B^2$

or

B = 340 feet

Length = $\frac{57,800}{B}$ = $\frac{57,800}{340}$

or

Length, L = 170

Hence, to minimize the cost the dimensions should be 340 feet on sides and 170 on the opposite side of the bank