To represent the total cost for a customer purchasing \( w \) car washes, we can use the following expression:
\[ \text{Total Cost} = 14w + 5 \]
Here, \( 14w \) represents the cost for the car washes (with \( w \) being the number of washes at a rate of $14 per wash), and \( 5 \) is the one-time fee for supplies.
Now, to find the total cost for 10 washes, we substitute \( w \) with 10:
\[ \text{Total Cost} = 14(10) + 5 \]
Calculating that:
\[ \text{Total Cost} = 140 + 5 = 145 \]
So, the total cost for 10 washes is $145.