To write a function for the length given the width of the rectangular play area, we can use the formula mentioned: 2(l + w) = 64.
Let's solve this equation for l:
2(l + w) = 64
Divide both sides by 2:
l + w = 32
Subtract w from both sides:
l = 32 - w
Therefore, the function for the length (l) given the width (w) is:
l = 32 - w
To graph this function, we will plot the length (l) on the y-axis and the width (w) on the x-axis. The graph will be a straight line with a negative slope, intersecting the y-axis at 32.
As for the reasonable domain for this situation, we need to consider the constraints of the problem. In this case, the width of the play area cannot be negative or greater than 32 (as the length cannot be negative). Additionally, since we are dealing with a physical object (fencing), we can assume that both the length and width should be positive values. Therefore, the reasonable domain for this situation would be 0 < w ≤ 32.