Answer:
Explanation:
To solve this problem, we can follow these steps:
Let's denote the dimensions of the rectangular area as length (L) and width (W) in feet.
The rancher has 750 feet of fencing, which will be used to enclose the perimeter of the rectangular area as well as to create divisions between the four pens.
The perimeter of the rectangular area is equal to the sum of all sides, which is given by:
Perimeter = 2L + 2W
The fencing for the divisions between the four pens will run the length of the rectangle (parallel to one side). Since there will be three divisions, the total length of fencing required for the divisions is 3L.
The remaining fencing is used to enclose the perimeter of the entire rectangular area, which is 750 - 3L.
Now, we can write an equation to represent the total available fencing equal to the sum of the fencing for divisions and the remaining fencing for the perimeter:
750 - 3L = 2L + 2W
Rearrange the equation to solve for W:
2W = 750 - 3L - 2L
2W = 750 - 5L
W = (750 - 5L)/2
We need to maximize the area of the rectangular enclosure, which is given by:
Area = L * W
Substitute the expression for W from step 7 into the area formula:
Area = L * [(750 - 5L)/2]
Simplify the equation:
Area = (750L - 5L^2)/2
To find the maximum area, we can take the derivative of the area function with respect to L and set it equal to zero:
d(Area)/dL = 750/2 - 10L/2 = 375 - 5L
Set the derivative equal to zero and solve for L:
375 - 5L = 0
5L = 375
L = 375/5
L = 75 feet
Now that we have found the value of L, we can find the corresponding value of W using the equation from step 7:
W = (750 - 5L)/2
W = (750 - 5(75))/2
W = (750 - 375)/2
W = 375/2
W = 187.5 feet
So, the dimensions of the rectangular enclosure that will maximize the area are:
Length (L) = 75 feet
Width (W) = 187.5 feet
Now, you can calculate the maximum area by multiplying these dimensions:
Maximum Area = L * W = 75 feet * 187.5 feet = 14,062.5 square feet