Question

The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs ₹ 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to ₹ 300 per hour is

Options

• 10

• 20

• 30

• 40

Answer 1

40

Explanation:

Let the speed of the train be v and distance to be covered be so that total time taken is s/v hours. Cost of fuel per hour = kv² (k is constant)

Also 48 = k. 16² by given condition

∴ k = `3/16`

∴ Cost to fuel per hour `3/16 v^2`

Other charges per hour are 300.

Total running, C = `(3/16 v^2 + 300) s/v = (3s)/16 v + (300s)/v`

`(dC)/(dv) = (3s)/16 - (300s)/v^2` = 0 ⇒ c = 40

`(d^2C)/(dv^2) = (600s)/v^3 > 0` 

∴ v = 40 results in minimum running cost