To express the relationship between the number of hours driven (\( h \)) and the total amount of miles driven (\( m \)), we can use the formula for distance:
\[ m = r \cdot h \]
where \( r \) is the rate of speed. In this case, the family is driving at a speed of 65 miles per hour. Thus, the equation relating \( h \) and \( m \) becomes:
\[ m = 65h \]
To find out how far they drove after 7 hours, we can substitute \( h = 7 \) into the equation:
\[ m = 65 \cdot 7 \]
Calculating that gives:
\[ m = 455 \]
So, after 7 hours, they drove a total of 455 miles.