Question

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

• A. $20$ m.p.h.
• B. $30$ m.p.h.
• C. $40$ m.p.h.
• D. $50$ m.p.h.

Answer 1

Answer: (C)

$40$ m.p.h.

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= $k{v}^2$ (k is constant)

Also $48=k{16}^2$ by given condition $\therefore k=3/16.$

$\therefore$ Cost of fuel per hour $=\frac{3}{16}{v}^2.$

Other charges per hour are 300.

$\therefore$ Charges per hour are $\frac{3}{16}{v}^2+300.$

$\therefore$ Total expenses for the journey

$E=(\frac{3}{16}{v}^2+300)\frac{S}{v}$

or $E=S(\frac{3}{16}v+\frac{300}{v})$

$\frac{dE}{dv}=S(\frac{3}{16}-\frac{300}{v^2})=0$

$\therefore v^2=1600orv=40$

$\frac{d^{2}E}{d{v}^2}=S\left(\frac{600}{v^3}\right)=+ive$ for v=40

and hence E is minimum.

Hence the most economical speed is 40 m.p.h.

Ans: C