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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Self Evaluation Test - Application of Derivatives

  • question_answer
    The cost of running a bus from A to B, is Rs. \[\left( av+\frac{b}{v} \right),\] where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/ h) of the bus is:

    A) 45

    B) 50

    C) 60

    D) 40

    Correct Answer: C

    Solution :

    [c] Let cost \[C=av+\frac{b}{v}\] According to given question, \[30a+\frac{b}{30}=75\]                                               ?(i) \[40a+\frac{b}{40}=65\]                                               ?(ii) On solving (i) and (ii), we get \[a=\frac{1}{2}\] and \[b=1800\] Now, \[C=av+\frac{b}{v}\Rightarrow \frac{dC}{dv}=a-\frac{b}{{{v}^{2}}}\] \[\frac{dC}{dv}=0\Rightarrow a-\frac{b}{{{v}^{2}}}=0\Rightarrow v=\sqrt{\frac{b}{a}}=\sqrt{3600}\] \[\Rightarrow v=60\,\,kmph\]


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