Applications Of Derivatives Question 62
Question: The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs Rs. 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to Rs. 300 per hour is
Options:
A) 10
B) 20
C) 30
D) 40
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Let the speed of the train be v and distance to be covered be s so that total time taken is s/v hours. Cost of fuel per hour $ =kv^{2} $ (k is constant) also $ 48=k.16^{2} $ by given condition
$ \therefore k=\frac{3}{16} $
$ \therefore $ Cost to fuel per hour $ \frac{3}{16}v^{2} $ . Other charges per hour are 300. Total running cost, $ C=( \frac{3}{16}v^{2}+300 )\frac{s}{v}=\frac{3s}{16}v+\frac{300s}{v} $
$ \frac{dC}{dv}=\frac{3s}{16}-\frac{300s}{v^{2}}=0\Rightarrow v=40 $
$ \frac{d^{2}C}{dv^{2}}=\frac{600s}{v^{3}}>0\therefore v=40 $ results in minimum running cost.
BETA
