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The fuel cost for running a train is proportional to the square of the speed generated in km per hour. If the fuel costs ₹ 48 per hour at speed 16 km per hour and the fixed charges amount to ₹ 1200 per hour then find the most economical speed of train, when total distance covered by train is S km.

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Let F be the fuel cost per hour and v the speed in km/h.

From question,

If t is the time taken by train in covering given distance S km and C the total cost for running the train then,

For maximum or minimum value of C, \(\frac{dC}{dv}=0\) 

Also,

⇒ \(\frac{d^2C}{dv^2}=\) 2400 x \(\frac{S}{v^2}\)

⇒ \((\frac{d^2C}{dv^2})_{v=80} > 0\) 

Hence, C is minimum, when v = 50 km/hour.

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