To represent the total cost for a customer who purchases \( w \) car washes, you can use the following expression:
\[ \text{Total Cost} = 10w + 2 \]
In this expression:
- \( 10w \) represents the cost of the car washes, where \( w \) is the number of car washes purchased (at $10 per wash).
- \( 2 \) is the one-time fee for supplies.
So, the total cost for a customer who buys \( w \) car washes is \( 10w + 2 \).