Question

(10) 6. you need to enclose four adjacent rectangular pens next to a river with total area 50,000 square feet. no fence is needed along the river. find the dimensions ( x ) and ( y ) that minimize the length of fence needed.

Explanation:

Step1: Set up the area and perimeter equations

The area of the four - adjacent rectangular pens is \(A = xy=50000\), so \(y=\frac{50000}{x}\).
The length of the fence \(L = 5x + y\) (since there are 5 sides of length \(x\) and 1 side of length \(y\) not along the river).

Step2: Substitute \(y\) into the fence - length equation

Substitute \(y=\frac{50000}{x}\) into \(L\): \(L(x)=5x+\frac{50000}{x}\), \(x>0\).

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

Using the power rule, if \(L(x)=5x + 50000x^{-1}\), then \(L^\prime(x)=5-\frac{50000}{x^{2}}\).

Step4: Set the derivative equal to zero and solve for \(x\)

Set \(L^\prime(x) = 0\):

$$
LATEXBLOCK0
$$

Step5: Check the second - derivative

Find the second - derivative \(L^{\prime\prime}(x)=\frac{100000}{x^{3}}\). When \(x = 100\), \(L^{\prime\prime}(100)=\frac{100000}{100^{3}}=\frac{1}{10}>0\), so \(L(x)\) has a minimum at \(x = 100\).

Step6: Find the value of \(y\)

Substitute \(x = 100\) into \(y=\frac{50000}{x}\), we get \(y = 500\).

Answer:

\(x = 100\) feet and \(y = 500\) feet.