Final answer:
To find the maximum area of the two adjacent rectangular pens with a total fence length of 300 ft, we need to first express the fence length in terms of the dimensions of the pens.
Step-by-step explanation:
To find the maximum area of the two adjacent rectangular pens, we first need to express the total amount of fence, which is 300 ft, in terms of the dimensions of the pens.
Let's assume the length of one pen is x ft and the other pen has a width of y ft. Since the two pens are adjacent, the total length of the shared fence would be 2y ft.
So, the total length of the fences for the two pens can be expressed as 2x + 2y = 300.
Next, we can rearrange the equation to y = (300 - 2x) / 2 = 150 - x. Now, to find the area of the pens, we multiply the length and width. The area of the two pens is given by A = x*y = x*(150 - x).
In order to find the maximum area, we can take the derivative of the area function with respect to x, set it equal to zero, and solve for x.
The maximum area occurs at a critical point, which is x = 75 ft. Substituting this value back into the equation y = 150 - x, we find y = 150 - 75 = 75 ft.
Therefore, the maximum area that can be enclosed is A = 75*75 = 5625 ft2.