Question

starting with a 140 - foot - long stone wall, a farmer would like to construct a rectangular enclosure by adding 700 feet of fencing, as shown in the figure to the right. find the values of x and w that result in the greatest possible area.
x =
w =

Explanation:

Step1: Set up the perimeter equation

The total length of the fencing is 700 feet. The perimeter equation considering the stone - wall is \(2w+(x + 140)=700\), which simplifies to \(2w+x=560\), and then \(x = 560 - 2w\).

Step2: Set up the area equation

The area of a rectangle \(A=(x + 140)w\). Substitute \(x = 560 - 2w\) into the area formula: \(A=(560 - 2w+140)w=(700 - 2w)w=700w-2w^{2}\).

Step3: Find the derivative of the area function

Differentiate \(A(w)=700w - 2w^{2}\) with respect to \(w\). Using the power rule \((x^n)^\prime=nx^{n - 1}\), we get \(A^\prime(w)=700-4w\).

Step4: Find the critical points

Set \(A^\prime(w)=0\), so \(700 - 4w=0\). Solving for \(w\) gives \(4w=700\), and \(w = 175\) feet.

Step5: Find the value of \(x\)

Substitute \(w = 175\) into the equation \(x = 560 - 2w\). Then \(x=560-2\times175=560 - 350 = 210\) feet.

Answer:

\(x = 210\), \(w = 175\)