Find the speed that minimizes the cost per mile.

Question

The cost of fuel to run a locomotive is proportional to the square of the speed and is $25/h for a speed of 25 mi/h. Other costs amount to $100/h, regardless of the speed. Find the speed that minimizes the cost per mile.

Answer 1

Let:

C = the cost per hour

v = the speed of a locomotive

The cost of fuel to run a locomotive is proportional to the square of the speed:

C = kv²

If C = $25/h then v = 25mi/h

25 = k(25)²

k= 1/25

C= v²/25

Other cost amount to $100/h regardless of speed.

C= v²/25 + 100/v

dC/dv = 2v/25 -100/v²

To get the minimum cost, set dC/dv = 0

0= 2v/25 -100/v²

0 = (2v³ - 2500)/25v²

0 = 2v³ - 2500

-2v³ = -2500

v³ = 1250

v ≈ 10.77

Therefore, the speed that minimizes the cost per mile:

C= (10.77)²/25 + 100/10.77

C≈ $13.93/ mile