g The radius of a spherical ball increases at a rate of 3 m/s. At what rate is the volume changing when the radius is equal to 2 meters

Question

g The radius of a spherical ball increases at a rate of 3 m/s. At what rate is the volume changing when the radius is equal to 2 meters

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Ngọc Khuê 1 year 2021-08-21T07:36:51+00:00 1 Answers 3 views 0

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    2021-08-21T07:37:56+00:00

    Answer:

    dV/dt = 150.79 m^3/s

    Explanation:

    In order to calculate the rate of change of the volume, you calculate the derivative, respect to the radius of the sphere, of the volume of the sphere, as follow:

    [tex]\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)[/tex]                     (1)

    r: radius of the sphere

    You calculate the derivative of the equation (1):

    [tex]\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)=3\frac{4}{3}\pi r^2\frac{dr}{dt}=4\pi r^2\frac{dr}{dt}\\\\\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}[/tex](2)

    where dr/dt = 3m/s

    You replace the values of dr/dt and r=2m in the equation (2):

    [tex]\frac{dV}{dt}=4\pi (2m)^2(3\frac{m}{s})=150.79\frac{m^3}{s}[/tex]

    The rate of change of the sphere, when it has a radius of 2m, is 150.79m^3/s

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