Fuel, Reserve, and Arrival Time on a 96-Mile Route
Officer Bhatt begins a shift in a patrol sedan with a 15-gallon tank that reads 3/8 full. The planned route is 96 miles. Department policy adds 8 percent to any mileage estimate to account for idling and detours. The sedan averages 17 miles per gallon. Policy also requires that the tank hold at least one quarter of its capacity when the vehicle returns to the station. Fuel is available only at the station, and the next scheduled refueling window opens after the shift ends. Traffic on the route runs steadily, so the average speed of 48 miles per hour is realistic.
1. Using the policy-adjusted mileage, determine whether the sedan can complete the route without refueling and still satisfy the quarter-tank reserve. Show each step.
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Step 1 - policy-adjusted distance 96 mi × 1.08 = 103.68 mi
Step 2 - fuel consumed 103.68 ÷ 17 = 6.099 gal (about 6.10 gal)
Step 3 - fuel on board at start 15 × 3/8 = 5.625 gal
Step 4 - fuel remaining on return 5.625 − 6.10 = −0.475 gal
The vehicle runs dry before completing the route. It falls short of the empty tank by about half a gallon, so it fails the reserve requirement by far more than that.
Step 5 - what the reserve actually demands Reserve = 15 × 1/4 = 3.75 gal Fuel needed at start = 6.10 + 3.75 = 9.85 gal Shortfall = 9.85 − 5.625 = 4.225 gal
Conclusion: the route cannot be run as planned. At least 4.23 gallons must be added before departure - call it 4.5 gallons to leave a working margin. Note that answering only "it runs out of gas" understates the problem: even a topped-off tank at exactly 6.10 gallons would complete the trip and still violate policy.
2. The officer must reach the far end of the route by 1430. With the two scheduled 12-minute stops and a 48 mph average, at what time must the officer depart? Round sensibly and state the margin you would build in.
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Step 1 - driving time Use the policy-adjusted 103.68 mi (the 8 percent allowance covers idling and detours, which consume time as well as fuel): 103.68 ÷ 48 = 2.16 hr = 2 hr + (0.16 × 60) = 2 hr 9.6 min, i.e. about 2 hr 10 min.
Step 2 - stops 2 × 12 min = 24 min
Step 3 - total 2 hr 10 min + 24 min = 2 hr 34 min
Step 4 - latest departure 1430 − 2 hr 34 min = 1156
Conclusion: the mathematical latest departure is 1156. I would brief a departure of 1150 - six minutes of margin absorbs one red light or a slower-than-average stretch, and a required arrival time is a deadline the officer cannot negotiate after the fact. Building the margin into the departure is cheaper than building it into the speed.
3. Suppose the officer instead departs with exactly 9.85 gallons, the minimum that satisfies policy. If traffic reduces the average speed to 40 mph, does the fuel plan still hold, and what is the new arrival estimate for the same 1430 deadline?
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Fuel check: the fuel plan is set by distance, not speed. The 8 percent allowance already covers idling and detours, and slower traffic does not change the route length unless the officer is rerouted. So 9.85 gallons still covers 103.68 miles and still leaves the 3.75-gallon reserve - the fuel plan holds, but with essentially zero slack. Any meaningful reroute would break it, since 9.85 − 3.75 = 6.10 gallons = 103.68 miles exactly.
Time check: 103.68 ÷ 40 = 2.592 hr = 2 hr + (0.592 × 60) = 2 hr 35.5 min Plus 24 min of stops = 2 hr 59.5 min, about 3 hr. Departing at 1156 now produces an arrival near 1456 - roughly 26 minutes late.
What this shows: the departure time computed at 48 mph has almost no cushion. To hold a 1430 arrival at 40 mph, departure must be no later than 1430 − 3 hr = 1130. The practical lesson is that the fuel margin and the time margin have to be computed separately; a plan that is tight on one is not rescued by the other.