a fence must be built to enclose a rectangular area of 20,000 ft². fencing material costs $3 per foot for…

a fence must be built to enclose a rectangular area of 20,000 ft². fencing material costs $3 per foot for the two sides facing north and south and $6 per foot for the other two sides. find the cost of the least expensive fence. the cost of the least expensive fence is $2400. (simplify your answer.)

a fence must be built to enclose a rectangular area of 20,000 ft². fencing material costs $3 per foot for the two sides facing north and south and $6 per foot for the other two sides. find the cost of the least expensive fence. the cost of the least expensive fence is $2400. (simplify your answer.)

Answer

Explanation:

Step1: Let the length of the north - south sides be (x) and the length of the other two sides be (y)

The area (A = xy=20000), so (y=\frac{20000}{x}) The cost function (C = 3\times2x+6\times2y=6x + 12y) Substitute (y=\frac{20000}{x}) into the cost function: (C(x)=6x+12\times\frac{20000}{x}=6x+\frac{240000}{x})

Step2: Find the derivative of the cost function

(C^\prime(x)=6-\frac{240000}{x^{2}}) Set (C^\prime(x) = 0) (6-\frac{240000}{x^{2}}=0) (6=\frac{240000}{x^{2}}) (x^{2}=\frac{240000}{6} = 40000) (x = 200) (we take (x>0) since (x) represents a length)

Step3: Find (y)

When (x = 200), (y=\frac{20000}{200}=100)

Step4: Calculate the cost

(C=6\times200+12\times100=1200 + 1200=2400)

Answer:

The cost of the least expensive fence is ($2400)