One side of a rectangular animal pen is to be formed by the side of a barn. If you have enough material to build a total of 34 feet of fence, find the maximum area than can be enclosed
Let side x be parallel to the barn. Then
x+2y = 34
a = xy = (34-2y)y = 34y-2y^2
da/dy = 34-4y
max area when da/dy=0, or y=17/2
So, the pen is 17 by 17/2
As usual, max area is when the fencing is divided equally among lengths and widths.
To find the maximum area that can be enclosed, we need to determine the dimensions of the rectangular animal pen that will maximize the area. Let's assume the length of the pen is x feet, and the width is y feet.
Since one side of the pen is formed by the side of a barn, we can determine that the width of the pen is equal to y, and the remaining three sides will contribute to the total length, which will be x + x + y.
Given that we have enough material to build a total of 34 feet of fence, we can create an equation based on the perimeter of the pen:
2x + y = 34
To find the dimensions that maximize the area, we need to express the area of the pen, A, in terms of x and y. The area of a rectangle is given by A = length × width, so the area of the pen will be A = (x + x + y) × y = (2x + y) × y.
Now we'll substitute the perimeter equation into the area equation to express the area solely in terms of x:
A = (2x + y) × y
A = (34 - y) × y
To maximize the area, we'll take the derivative of A with respect to y and set it equal to zero:
dA/dy = (34 - y) - y = 34 - 2y
Setting dA/dy to zero:
34 - 2y = 0
2y = 34
y = 17
Now that we have the value of y, we can substitute it back into our perimeter equation to solve for x:
2x + 17 = 34
2x = 17
x = 8.5
So, the maximum area that can be enclosed is achieved when the length (x) is 8.5 feet and the width (y) is 17 feet.