Let's break down the problem step by step:

The dog run has a rectangular shape, with one of the shorter sides (7.5 feet) formed by the shelter building. The volunteers are fencing the other sides of the rectangle, so the total perimeter of the fencing includes the two long sides (both of length $x$) and one short side (7.5 feet).

The cost of fencing is $2.50 per foot, and the total cost is limited to between $100 and $200.

Step 1: Set up the perimeter equation

The total length of fencing required is for the two long sides and one short side:

$\text{Total length of fencing} = 2x + 7.5$

Step 2: Set up the cost equation

At $2.50 per foot, the total cost of the fencing is:

$\text{Total cost} = 2.50 \times (2x + 7.5)$

Step 3: Write the compound inequality for cost

The volunteers want to spend between $100 and $200 on fencing. Therefore, the inequality is:

$100 \leq 2.50 \times (2x + 7.5) \leq 200$

Step 4: Solve the inequality

We'll now solve this inequality to find the possible values for $x$.

1. Start by dividing the entire inequality by 2.50 to simplify:

$\frac{100}{2.50} \leq 2x + 7.5 \leq \frac{200}{2.50}$

$40 \leq 2x + 7.5 \leq 80$

2. Subtract 7.5 from all parts of the inequality:

$40 - 7.5 \leq 2x \leq 80 - 7.5$

$32.5 \leq 2x \leq 72.5$

3. Finally, divide by 2:

$\frac{32.5}{2} \leq x \leq \frac{72.5}{2}$

$16.25 \leq x \leq 36.25$

Conclusion

The possible length of the dog run, $x$, can be between 16.25 feet and 36.25 feet.

Would you like further details on any step, or do you have any other questions?

Here are 5 related questions:

1. How would the cost change if the price per foot of fencing increased to $3.00?
2. What if the budget was expanded to $300 for the fencing? How would that affect the length?
3. How would the inequality change if both shorter sides of the rectangle needed to be fenced?
4. Could you solve for the length if the cost limit was only $150?
5. How much total fencing would be needed if the length was exactly 30 feet?

Tip: Always double-check units and make sure to maintain consistency throughout your calculations!