Question

The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹ 48 per hour at speed 16 km per hour and the fixed charges to run the train amount to ₹ 1200 per hour. Assume the speed of the train as v km/h.

Based on the given information, answer the following questions.

(i) Given that the fuel cost per hour is times the square of the speed the train generates in km/h, the value of k is :

(a) 16/3

(b) 1/3

(c) 3

(d) 3/16

(ii). If the train has travelled a distance of 500 km, then the total cost of running the train is given by function :

(a) \(\frac{15}{16}v+\frac{600000}{v}\) 

(b) \(\frac{375}{4}v+\frac{600000}{v}\)

(c) \(\frac{15}{16}v^2+\frac{150000}{v}\)

(d) \(\frac{3}{16}v+\frac{6000}{v}\)

(iii). The most economical speed to run the train is :

(a) 18 km/h

(b) 5 km/h

(c) 80 km/h

(d) 40 km/h

(iv). The fuel cost for the train to travel 500 km at the most economical speed is :

(a) ₹ 3750

(b) ₹ 750

(c) ₹ 7500

(d) ₹ 75000

(v) The total cost of the train to travel 500 km at the most economical speed is :

(a) ₹ 3750

(b) ₹ 75000

(c) ₹ 7500

(d) ₹ 15000

Answer 1

(i). Option : (d)

Fuel cost= k(speed)²

⇒ 48 = k.16²

⇒ k = \(\frac{3}{16}\)

(ii) Option : (b)

Let the total cost of running train is C.

According to the question,

\(\frac{dC}{at}=\frac{3}{16}v^2+1200\)

\(\Rightarrow C=\frac{3}{16}v^2t+1200t\) (By integrating both sides w.r.t t)

Hence, total cost of running train is \(C=\frac{3}{16}\nu^2t+1200t\)

Distance covered = 500km

\(\Rightarrow\) time \(=\frac{500}{\nu}hrs\)

Total cost of running train 500 km \(=\frac{3}{16}\nu^2\left(\frac{500}{\nu}\right)+1200\left(\frac{500}{\nu}\right)\)

\(\Rightarrow C=\frac{375}{4}\nu+\frac{600000}{\nu}\)

(iii). Option : (c)

\(\frac{dC}{dv}\) = \(\frac{375}{4}\) - \(\frac{600000}{v^2}\)

Let \(\frac{dc}{dv}\) = 0

⇒ v² = \(\frac{600000\times 4}{375}\)= 6400

⇒ v = 80 km/h

(iv). Option : (c)

Fuel cost for running train to travel 500 km at the most economic speed is \(\frac{375}{4}v\) 

= \(\frac{375}{4}\) x 80

= Rs. 7500/−

(v). Option : (d)

Total cost for running 500 km = \(\frac{375}{4}v\) + \(\frac{600000}{v}\)

= \(\frac{375\times 80}{4}\)+\(\frac{600000}{v}\)

= Rs.15000/−