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Thread: Volume of a Ball

  1. #1
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    Volume of a Ball

    I need help with this problem:
    The radius of a spherical balloon is increasing by 2 cm/sec. At what rate is air being blown into the balloon when the radius is 10 cm?
    So far what I have is that the volume of the ball is $\displaystyle \frac4 3\pi r^3$ and that the radius $\displaystyle r=2t$ where $\displaystyle t$ is the time in seconds after the balloon began to inflate. Then the substitution could be made that the volume is $\displaystyle \frac4 3\pi (2t)^3 = 8\pi t^3$. Then to find the rate of change, take the derivative, giving $\displaystyle 24\pi t^2$. Then the radius will be 10 cm when t=5 so I plug in 5 for t, giving $\displaystyle 24*25*\pi = 600\pi$ $\displaystyle cm^3/min$. This answer feels wrong to me, as the total volume of the balloon is only about 4188.79 cubic cm. Could somebody point out where I went wrong if I have done so, thanks.
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  2. #2
    MHF Contributor
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    Pauls Online Notes : Calculus I - Related Rates

    If that didn't help:

    $\displaystyle \displaystyle V=\frac{4}{3}\pi r^3 \ \mbox{and} \ \frac{dr}{dt}=2cm/sec$

    Derivative of V $\displaystyle \displaystyle \frac{dV}{dt}=4*\pi r^2\frac{dr}{dt}=4*\pi*10^2*2=.....$
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