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 Rs 48 per hour at speed 16 km per hour and the fixed charges to run the train amount to Rs1200 per hour. Assume the speed of the train as v km/h.
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(i) Given that the fuel cost per hour is k times the square of the speed the train generates in km/h, the value of k is:
(a) \( \dfrac{16}{3}\)
(b) \( \dfrac{1}{3}\)
(c) 3
(d) \( \dfrac{3}{16}\)
(ii) If the train has travelled a distance of 500km, then the total cost of running the train is given by function:
(a) \( \dfrac{15}{16}v+\dfrac{600000}{v}\)
(b) \( \dfrac{375}{4}v+\dfrac{600000}{v}\)
(c) \( \dfrac{5}{16}v^2+\dfrac{150000}{v}\)
(d) \( \dfrac{3}{16}v+\dfrac{6000}{v}\)
(iii) The most economical speed to run the train is:
(a) 18km/h
(b) 5km/h
(c) 80km/h
(d) 40km/h
(iv) The fuel cost for the train to travel 500km at the most economical speed is:
(a) Rs 3750
(b) Rs 750
(c) Rs 7500
(d) Rs 75000
(v) The total cost of the train to travel 500km at the most economical speed is:
(a) Rs 3750
(b) Rs 75000
(c) Rs 7500
(d) Rs 15000
1 Answer
Solution :-
(i) Right answer is (d) \( \dfrac{3}{16}\)
Explanation:-
Fuel cost = \( k(speed)^2\)
= 48 =k.162
k =\( \dfrac{3}{16}\)
(ii) Right answer is (b) \( \dfrac{375}{4}v+\dfrac{600000}{v}\)
Explanation:-
Total cost of running train (let C) = \( \dfrac{3}{16}v^2t+1200t\)
Distance covered = 500km
time=\( \dfrac{500}{v}hrs\)
Total cost of running train 500 km
=\( \dfrac{3}{16}v^2(\dfrac{500}{v})+1200(\dfrac{500}{v})\)
= \( \dfrac{375}{4}v+\dfrac{600000}{v}\)
(iii) Right answer is (c) 80km/h
Explanation:-
\( \dfrac{dc}{dv}=\dfrac{375}{4}-\dfrac{600000}{v^2}\)
Let \( \dfrac{dc}{dv}=0\)
v = 80km/h
(iv) Right answer is (c) Rs 7500
Explanation:-
Fuel cost for running 500 km
= \( \dfrac{375}{4}v=\dfrac{375}{4}\times 80=Rs. 7500\)
(v) Right answer is (d) Rs 15000
Explanation:-
Total cost for running 500 km
=\( \dfrac{375}{4}v+\dfrac{600000}{v}\)
= \( \dfrac{375\times 80}{4} +\dfrac{600000}{80}\)
= Rs.15000
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