Question 2 OR (b)
Most economical speed of an engine
9 marks
- Time to write
- About 18 minutes
Question
Question 2 OR (b) · 9 marksThe cost of fuel for running an engine varies as the square of the speed in kms per hour, and is ₹36 per hour at a speed of 20 kms. Other costs amount to ₹225 per hour. Find the most economical speed.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Fuel cost function (k) 2 marks 4 min
- 2 Cost per km 2 marks 3 min
- 3 First- and second-order conditions 4 marks 7 min
- 4 Economical speed and cost 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Cost per hour. Fuel cost per hour varies as the square of the speed v (km per hour): fuel = kv². At v = 20 it is ₹36, so 36 = 400k ⟹ k = 0.09. Total cost per hour = 0.09v² + 225.
Cost per kilometre. In one hour the engine covers v km, so the cost per km is
C(v) = (0.09v² + 225)/v = 0.09v + 225/v.
(The economical speed minimises the cost of covering a given distance, i.e. the cost per km, not the cost per hour.)
dC/dv = 0.09 − 225/v² = 0 ⟹ v² = 2,500 ⟹ v = 50 km per hour. d²C/dv² = 450/v³ > 0 for v > 0, so this is a minimum (derivatives by the power rule).
At v = 50: fuel cost 0.09 × 2,500 = ₹225 an hour; other costs ₹225 an hour; cost per km = 450/50 = ₹9.
| Speed (km/h) | 40 | 50 | 60 |
|---|---|---|---|
| Fuel cost per hour (₹) | 144 | 225 | 324 |
| Cost per km (₹) | 9.225 | 9.000 | 9.150 |
The most economical speed is 50 km per hour (at that speed fuel cost per hour equals the other costs per hour).
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Fuel cost function (k) | 2 |
| 2 | Cost per km | 2 |
| 3 | First- and second-order conditions | 4 |
| 4 | Economical speed and cost | 1 |
| Total | 9 |
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