Final answer:
The area of a playground with one side against a building and 3 sides fenced with 200 meters of fencing is 5000 square meters when x, the length of each side perpendicular to the building, is 50 meters. a reasonable domain for A is 0 < x ≤ 100 meters.
Step-by-step explanation:
To find the area of the playground when x = 50 meters, we recall that the total amount of fencing available is 200 meters. Since the fencing is used for three sides, and one side (width) x is given, we can determine the other side (length) by subtracting twice the width from the total fencing, as the remaining two sides are parallel to each other and equal in length. This gives us a length of 200 - 2x. Therefore, at x = 50 meters, the length of the playground is 200 - 2(50) = 100 meters.
The area A(x) can thus be expressed as A(x) = x × (200 - 2x). The area of the playground when x = 50 is then A(50) = 50 × (200 - 2 × 50) = 50 × 100 = 5000 square meters.
Considering the reasonable domain for A(x), x must be greater than 0 but less than or equal to 100 since the total fencing is 200 meters and we need to allow for the other two sides of fencing. Thus, a reasonable domain for A is 0 < x ≤ 100 meters.