Explanation:

To solve this problem:

- Let's assume the distance between the hilltop and the foothill is \(d\) miles.

- Shreya hikes up at 5 mph and comes down at 7 mph.

- Time taken to hike up is \(\frac{d}{5}\) hours.

- Time taken to come down is \(\frac{d}{7}\) hours.

- Total time for the journey is 3 hours.

We can set up the equation:

$$

\frac{d}{5} + \frac{d}{7} = 3

$$

- Multiply the entire equation by 35 (LCM of 5 and 7) to clear fractions:

$$

7d + 5d = 105

$$

- Combine like terms:

$$

12d = 105

$$

- Solve for \(d\):

$$

d = \frac{105}{12} = 8.75 \text{ miles}

$$

- Therefore, the correct option is: Option 1: 8.75 miles