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Math Help - Related Rates about a water tank the shape of a sphere

  1. #1
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    Related Rates about a water tank the shape of a sphere

    A water tank that is in the shape of a sphere has a radius of 4 meters. When the tank is filled to a depth of
    h meters, the volume of water in the tank is

    V
    =p/3 h^2(12 h).

    h is the height of water, measured in meters from the bottom of the tank.

    *What values of
    h are possible.
    *Suppose that the tank is initially empty but that water is poured into the tank at the constant rate of 2 cubic meters per minute. What is the total volume of the tank? How long does it take to fill the tank?

    *Determine a formula for the rate of change of the height of thewater with respect to time. Your formula should only depend on
    h.

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  2. #2
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    The total volume of a sphere is \frac{4}{3}{\pi}r^{3}

    Therefore, the volume of a sphere with radius 4 meters is \frac{256\pi}{3} cubic meters.

    If the water flows in at 2 cubic meters per minute, then it takes:

    \frac{256\pi}{6}=134.04 minutes.

    Can you use related rates to find dh/dt?. Differentiate your formula wrt time.

    \frac{dV}{dt}={\pi}\left(8h-h^{2}\right)\cdot \frac{dh}{dt}

    Solve for dh/dt. You know dV/dt=2
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  3. #3
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    Related rates reply

    Thank you so much. We can use related rates to find dh/dt. I'm confused as to how you found the equation for dV/dt?
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  4. #4
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    I merely differentiated, wrt h, the volume formula you were given.

    But h varies with time, so we have dh/dt. The change in h as time progresses.
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