Final answer:
a) The function that gives the area of the rectangular field as a function of x is A(x) = x * (140 - x). b) The side length x that gives the maximum area is 70. c) The maximum area that the field can have is 4900 square meters.
Step-by-step explanation:
a) To find the function that gives the area of the rectangular field in terms of x, we need to use the perimeter equation to solve for one of the side lengths. The perimeter of a rectangle is given by the equation P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. In this case, we have a total of 280 meters of fencing, so we can write the equation as 280 = 2L + 2W. Since the length is x, we can substitute L with x and solve for W: 280 = 2x + 2W. Subtracting 2x from both sides gives us 280 - 2x = 2W. Dividing by 2 gives us W = 140 - x. Now, we can use the area equation A = L * W to find the area as a function of x: A(x) = x * (140 - x).
b) To find the side length x that gives the maximum area, we can take the derivative of A(x) with respect to x and set it equal to zero. The derivative of A(x) is dA/dx = 140 - 2x. Setting this equal to zero and solving for x gives us 140 - 2x = 0, which simplifies to x = 70. So, the side length x that gives the maximum area is 70.
c) To find the maximum area, we can substitute x = 70 into the area function A(x): A(70) = 70 * (140 - 70) = 70 * 70 = 4900 square meters. Therefore, the maximum area that the field can have is 4900 square meters.