The Range of Taxi Fares

You are in the back of a NYC taxi, and know exactly how fare your trip was in miles (m), and exactly how long it took in hours (t). The taxi driver gives you bill, but you want to make sure they’ve charged you correctly.

You know that taxi fares are calculated as follows:
"There is a $2.50 initial charge, plus $2.50 cents per mile when traveling above 12mph or $0.50 per minute in slow traffic or when the vehicle is stopped."

You know that your taxi never traveled above 60mph, but can’t recall what proportion (p) of time was spent traveling above 12mph.

The question is, what is the minimum and maximum amount (in terms of t and m) that your fare should cost?

For example, if the trip was 10 miles in 1 hour, it could have been 50 mins stationary and 10 mins @ 60mph (This would be 2.5 + 0.5*50 + 2.5*10 = $52.50). And this is not as simple as assuming p=0 and p=1 are the bounds.