a rancher has 12,000 feet of fence to enclose 5 adjacent pens. determine the outer rectangular dimensions that result the maximum enclosed area.
To determine the outer rectangular dimensions that result in the maximum enclosed area, we need to use the concept of calculus, specifically optimization. The problem is essentially finding the dimensions of a rectangle that maximizes its area while having a fixed perimeter.
Let's denote the length of the rectangle as L and the width as W. Given that there are five adjacent pens, the total length of all the sides that will be enclosed by the fence is 5L.
Now, we know that the perimeter of the rectangle is equal to the total length of the fence available, which is 12,000 feet:
Perimeter = 2L + 2W = 12,000
We can simplify this to:
L + W = 6,000
To find the dimensions that maximize the area, we need to express the area in terms of a single variable. The area of a rectangle is given by:
Area = Length × Width = L × W
Since we have already expressed L in terms of W (L = 6,000 - W), we can substitute it into the area equation:
Area = (6,000 - W) × W = 6,000W - W^2
Now, we have an equation for the area in terms of a single variable, W. To find the maximum area, we need to find the value of W that maximizes this equation.
To maximize the area, we can take the derivative of the area equation with respect to W and set it equal to zero:
d(Area)/dW = 6,000 - 2W = 0
Solve for W:
6,000 - 2W = 0
2W = 6,000
W = 3,000
Now, substitute this value of W back into the equation for the perimeter to find the corresponding value of L:
L + W = 6,000
L + 3,000 = 6,000
L = 6,000 - 3,000
L = 3,000
So, the dimensions of the rectangle that results in the maximum enclosed area are L = 3,000 feet and W = 3,000 feet.