Yolanda has 400 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not need fencing.

Question

As shown below, on of the sides has length x ( in meters).

1 find a function that gives the area A(x) of the field (in square meters) in terms of x

A(x)=_____

2 What side length x gives the maximum area that the field can have?

side length x: _____meters

3 what is the maximum area that the field can have?

maximum area______ square meters

Answer 1

1. Assuming that x is the length of the side that parallels the river, each of the two sides perpendicular to the river will be (400-x)/2 long. (x amount of the 400 m of fence will be used for the one parallel side, leaving 400-x to be divided among the 2 perpendicular sides) Therefore, A(x) = x(400-x)/2. If, instead, x was the length of one of the sides perpendicular to the river, the parallel side would be 400-2x and A(x) = x(400-2x)
2. Assuming the first scenario, we want to maximize x(400-x). This will occur when x = 400-x or at x=200. In the 2nd scenario, the maximum occurs when x = 400-2x or at x=133 1/3
3. In the first scenario, 20,000 m², in the 2nd 17,777.7... m²