Question

a fence must be built to enclose a rectangular area of 5000 ft². fencing material costs $1 per foot for the two sides facing north and south and $2 per foot for the other two sides. find the cost of the least expensive fence. the cost of the least expensive fence is $□. (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 \(A = xy=5000\), so \(y=\frac{5000}{x}\). The cost function \(C = 1\times2x+2\times2y=2x + 4y\)

Step2: Substitute \(y=\frac{5000}{x}\) into the cost function

\(C(x)=2x + 4\times\frac{5000}{x}=2x+\frac{20000}{x}\)

Step3: Find the derivative of \(C(x)\)

Using the power rule \((x^n)^\prime=nx^{n - 1}\), \(C^\prime(x)=2-\frac{20000}{x^{2}}\)

Step4: Set \(C^\prime(x) = 0\) to find the critical points

\(2-\frac{20000}{x^{2}}=0\)
\(2=\frac{20000}{x^{2}}\)
\(x^{2}=10000\)
\(x = 100\) (we take \(x>0\) since \(x\) represents a length)

Step5: Find \(y\) when \(x = 100\)

Since \(y=\frac{5000}{x}\), then \(y = 50\)

Step6: Calculate the cost

\(C=2x+4y\), substitute \(x = 100\) and \(y = 50\)
\(C=2\times100+4\times50=200 + 200=400\)

Answer:

\(400\)