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

Answer 2

The dimensions that will minimize costs 170 feet on the side opposite the river and 340 feet on the other side.

How to  solve an equation?

Let x represent the side opposite the river and y represent the other side.

The field must contain 57,800 square feet, hence:

57800 = xy

y = 57800/x

The total cost is:

Cost (C) = 40x + 10y + 10y

C = 40x + 20y

C = 40x + 20(57800/x)

The minimum cost is at C'(x) = 0; hence:

C'(x) = 40 - 1156000/x²

0 = 40 - 1156000/x²

40 = 1156000/x²

40x² = 1156000

x = 170 feet

y = 57800/x = 57800/170 = 340 feet

The dimensions that will minimize costs 170 feet on the side opposite the river and 340 feet on the other side.

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