Final answer:
To express the area of the rectangular lot as a function of the length of the dividing fence, we need to determine the dimensions of the lot. By using the perimeter equation and substituting the expression for length into the area equation, we find that A(x) = 100x - 0.25x^2.
Step-by-step explanation:
To express the area of the rectangular lot, A, as a function of the length of the fence that divides the rectangular lot, x, we need to determine the dimensions of the rectangular lot in terms of x. Let's assume the length of the rectangular lot is L and the width is W. The perimeter of the rectangular lot is equal to the sum of the lengths of all four sides, which can be expressed as:
P = 2L + 2W + x
We know that the perimeter of the rectangular lot is equal to 400 feet, so we can write the equation:
2L + 2W + x = 400
Simplifying this equation, we get:
2L + 2W = 400 - x
Dividing by 2 on both sides, we have:
L + W = 200 - 0.5x
The area of the rectangular lot can be expressed as A = L * W. We can solve for L in terms of W using the equation above:
L = 200 - 0.5x - W
Substituting this expression for L into the equation for the area, we get:
A(x) = (200 - 0.5x - W) * W
Expanding and simplifying, we find:
A(x) = 200W - 0.5xW - W^2
This expression represents the area of the rectangular lot, A, as a function of the length of the fence that divides the lot, x. Therefore, the correct answer is option b) A(x) = 100x - 0.25x^2.