starting with a 140 - foot - long stone wall, a farmer would like to construct a rectangular enclosure by…
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.\nx = \n w =
Answer
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)