Question

A rectangular playground is surrounded by a chain link fence that costs $2 per linear foot on three sides and a fancier wooden fence that costs $6 per linear foot on the fourth side. The area of the playground is to be 500 square feet. Find the dimensions of the rectangle that minimize the total cost of the fencing

Answer 1

Answer:

31.6 feet by 15.8 feet.

Step-by-step explanation:

Area of a Rectangle = Length X Breadth = LB

The area of the playground is to be 500 square feet

Therefore:

LB=500

Perimeter of the Rectangle = 2(L+B)

The Chain link fence that costs $2 per linear foot on three sides and a fancier wooden fence that costs $6 per linear foot on the fourth side.

Cost = ${2(L+2B)+6L}=2L+4B+6L=8L+4B

Cost= $(8L+4B)

From: LB=500, B=500/L

Substitute B into 8L+4B

C(L) = $8L+4(\frac{500}{L} )=8L+\frac{2000}{L} =\frac{8L^2+2000}{L}$

C(L)=$\frac{8L^2+2000}{L}$

The minimum cost of the fencing occurs when the dimensions are minimum.

If we take the derivative of C(L)

$C^{'}(L)=\frac{8L^2-2000}{L^2}$

At $C^{'}(L)=0$

$\frac{8L^2-2000}{L^2}=0\\8L^2-2000=0\\8L^2=2000\\L^2=250\\L=15.8 ft$

Recall: B=500/L

$B=\frac{500}{15.8}=31.6 ft$

The dimensions that minimizes the total cost of the fencing are 31.6 feet by 15.8 feet.