To determine the number of miles \( x \) in Lydia's taxi ride, we can set up an equation based on the information given.
The total cost of the taxi fare can be expressed as:
\[ \text{total cost} = \text{pick-up fee} + (\text{cost per mile} \times \text{number of miles}) \]
From the problem, we know:
- The pick-up fee is \( 4.10 \) dollars.
- The cost per mile is \( 2.50 \) dollars.
- The total fare is \( 36.60 \) dollars.
We can write the equation as:
\[ 36.60 = 4.10 + 2.50x \]
To find \( x \), we will first isolate the term involving \( x \). Subtract \( 4.10 \) from both sides:
\[ 36.60 - 4.10 = 2.50x \]
Calculating the left side:
\[ 32.50 = 2.50x \]
Now, divide both sides by \( 2.50 \) to solve for \( x \):
\[ x = \frac{32.50}{2.50} \]
Calculating this gives:
\[ x = 13 \]
Thus, the number of miles in the taxi ride is \( \boxed{13} \).