Let us denote the length of the rectangle (the fence in this context) by 'l' and the width of the rectangle by 'w'. For any rectangle, the formula to calculate the perimeter 'P' is given as P = 2l + 2w (where l is the length and w is the width).
Given that we have 48 feet of fencing available to us, that would mean our perimeter 'P' is 48 feet. So, following the perimeter formula, we have:
2l + 2w = 48
We can re-arrange this formula by solving for 'w' to find all potential values of 'w' (the width) that would satisfy this equation. This gives us:
w = (48 - 2l) / 2
In the context of this problem, since 'w' represents a physical length, we need to ensure that the value of 'w' is greater than 0 (as a negative length wouldn't make sense in this context). So, we need to ensure that we follow this inequality:
0 < w <= (48 - 2l) / 2
This equation gives us the range of potential values for 'w' (the width of the pen), and by extension gives us the possible dimensions that our rectangular pen can take on given the 48 feet of fencing available.
So, the possible dimensions of the rectangle (or the dog pen in this case) are represented by the inequality 0 < w <= (48 - 2l) / 2. This means that the width of the pen could be any value greater than 0 but not exceeding the value given by (48 - 2l) divided by 2. The exact length and width values will depend on the specific value of 'l' (the length of the fence) that is chosen. For instance, if 'l' was chosen as 10 feet, then 'w' must be a value greater than 0 feet but no more than 14 feet (as given by (48 - 2*10) / 2).