Final answer:
To find the maximum area that can be enclosed with a dividing fence, we need to optimize the dimensions of the rectangular grazing area. Using calculus, we can find that the maximum area is 90000 square feet with dimensions of 300 feet by 300 feet.
Step-by-step explanation:
To find the maximum area that can be enclosed, we need to optimize the dimensions of the rectangular grazing area. Let's assume that the length of one side of the rectangular grazing area is x, then the dimensions of the grazing area would be x by 600-x (since the dividing fence splits it in half).
The perimeter of the grazing area can be represented as:
P = 2x + 2(600-x) = 1200 feet
Simplifying the equation, we get:
2x + 1200 - 2x = 1200
1200 = 1200
As a result, the perimeter of the grazing area is fixed at 1200 feet. Now, let's find the area of the grazing area:
A = x(600-x)
To find the maximum area, we can use calculus. The first step is to take the derivative of the area function with respect to x:
A' = 600 - 2x
Setting the derivative equal to zero and solving for x:
600 - 2x = 0
2x = 600
x = 300
Therefore, the dimensions of the grazing area that maximize the area are 300 feet by 300 feet. Substituting this value back into the area function, we can determine the maximum area:
A = 300(600-300) = 90000 square feet