The fuel charges for running a train are proportional to the square of the speed generated in km/h, and the cost is Rs. 48 at 16 km/h. If the fixed charges amount to Rs. 300/h, the most economical speed is 60 km/h (b) 40 km/h 48 km/h (d) 36 km/h

Options

A. 60 `km//h`
B. 40 `km//h`
C. 48 `km//h`
D. 36 `km//h`

Answer

2

Explanation

fuel charges `prop v^(2)` let f represetn fuel charges then
`f prop v^(2) or f=kv^(2)`
Given that f=Rs `48//h v=16 km//h` thus
`48=k(16)^(2) or k =(3)/(16)`
From (1) F=`(3v^(2))/(16)`
Let the train covers `lambda` km t hours then
`lambda =vt or t=(lambda)/(v)`
therefore Fuel charge in time `t=3/16v^(2)xx(lambda)/(v)=(3 v lambda)/(16)`
Thus total cost for running the train
`C=(3vlambda)/(16)+3000xx(lambda)/(v)`
`therefore(dC)/(dv)=(3lambda)/(16)-(300lambda)/(v^(2)) and (d^(2)C)/(dv^(2))=(600lambda)/(v^(3))`
For the maximum or minimum values of C
`(dC)/(dv) =0 or v=40 km//h`
Thus C is minimum when v = 40 km//h