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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Application in Mechanics and Rate Measurer

  • question_answer
    The volume V and depth x of water in a vessel are connected by the relation \[V=5x-\frac{{{x}^{2}}}{6}\]and the volume of water is increasing at the rate of \[5c{{m}^{3}}/\sec \], when \[x=2cm\]. The rate at which the depth of water is increasing, is

    A)            \[\frac{5}{18}cm/\sec \]

    B)            \[\frac{1}{4}cm/\sec \]

    C)            \[\frac{5}{16}cm/\sec \]

    D)            None of these

    Correct Answer: D

    Solution :

               \[V=5x-\frac{{{x}^{2}}}{6}\Rightarrow \frac{dV}{dt}=5\frac{dx}{dt}-\frac{x}{3}.\frac{dx}{dt}\]                    Þ \[\frac{dx}{dt}=\frac{\frac{dV}{dt}}{\left( 5-\frac{x}{3} \right)}\Rightarrow {{\left( \frac{dx}{dt} \right)}_{x=2}}=\frac{5}{5-\frac{2}{3}}=\frac{15}{13}cm/\sec \].


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