Question

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.)

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 \(\$2400\)