Two-Route Trip Cost With Time Value
A student is planning a 250-mile drive home for a long weekend and is comparing two routes. Route A is 250 miles: 20 miles are through construction at about 40 mph, and the remaining 230 miles are highway at about 70 mph. The car gets 22 miles per gallon in construction and 30 miles per gallon on the highway. Route A also has a $4 toll. Route B is 220 miles at an average 55 mph, gets 28 miles per gallon, and has no toll. Gas costs $3.50 per gallon. The student values driving time at $12 per hour and will travel alone unless a friend joins. The friend would split fuel and toll costs equally but would not change the time value. The student wants to compare total trip cost, including the value of time, and see whether carpooling changes the decision.
1. Compute driving time and fuel cost for each route.
Read the worked answer
Route A: construction time = 20/40 = 0.5 h; highway time = 230/70 ≈ 3.2857 h. Total time = 3.7857 h, or about 3 h 47 min. Fuel: construction = 20/22 ≈ 0.9091 gal; highway = 230/30 ≈ 7.6667 gal. Total fuel ≈ 8.5758 gal, costing 8.5758 × $3.50 ≈ $30.02.
Route B: time = 220/55 = 4.00 h. Fuel = 220/28 ≈ 7.8571 gal, costing 7.8571 × $3.50 ≈ $27.50.
Route A is faster by 4.00 - 3.7857 = 0.2143 h, or about 12.9 minutes, but it uses more fuel because the highway mileage is higher but the construction segment is inefficient and the route is longer overall.
2. Including the value of time and the toll, which route has the lower total cost when driving alone? Show the totals.
Read the worked answer
Route A: fuel $30.02 + toll $4.00 + time value (3.7857 × $12 = $45.43) = $79.45.
Route B: fuel $27.50 + time value (4.00 × $12 = $48.00) = $75.50.
Route B is lower by $79.45 - $75.50 = $3.95. The time saving on Route A is only 0.2143 h × $12 ≈ $2.57, while Route A’s fuel-plus-toll penalty is $34.02 - $27.50 = $6.52. The net disadvantage of Route A is $6.52 - $2.57 = $3.95, so the faster route is not the cheaper route once the toll and fuel are counted.
3. If the friend carpools and splits fuel and toll equally, recompute each total cost per person, keeping the full time value with the student. Does the choice change?
Read the worked answer
Route A: fuel plus toll = $30.02 + $4.00 = $34.02. Split equally gives $17.01 per person. Add the student’s time value of $45.43: total = $62.44.
Route B: fuel = $27.50. Split equally gives $13.75 per person. Add time value $48.00: total = $61.75.
Route B is still lower, but only by $62.44 - $61.75 = $0.69. Carpooling halves the fuel and toll difference, which narrows the gap substantially because Route A’s cost disadvantage was concentrated in fuel and toll. The decision does not change, but it becomes nearly a tie, so a small change in gas price or toll would flip it.