The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing when the diameter is 60 mm

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The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing when the diameter is 60 mm

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Bình An 1 week 2021-07-22T08:08:48+00:00 1 Answers 5 views 0

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    2021-07-22T08:10:02+00:00

    Answer:

    The volume is increasing at a rate of 33929.3 cubic millimeters per second.

    Step-by-step explanation:

    Volume of a sphere:

    The volume of a sphere of radius r is given by:

    V = \frac{4\pi r^3}{3}

    In this question:

    We have to derivate V and r implicitly in function of time, so:

    \frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}

    The radius of a sphere is increasing at a rate of 3 mm/s.

    This means that \frac{dr}{dt} = 3

    How fast is the volume increasing when the diameter is 60 mm?

    Radius is half the diameter, so r = 30. We have to find \frac{dV}{dt}. So

    \frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}

    \frac{dV}{dt} = 4\pi (30)^2(3) = 33929.3

    The volume is increasing at a rate of 33929.3 cubic millimeters per second.

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