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Math Help - Rate Question

  1. #1
    Member CalcGeek31's Avatar
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    Rate Question

    A liquid is going through a conical filter whose top diameter is the same as its height. It pours into a right circular cylinder whose radius is 20cm. How fast is the liquid in the cylinder rising in cm/min if the liquid in the filter is 10 cm high and falling at a rate of 80cm/min?


    I am not even sure where to start this question, any tips/help would be appreciated.
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  2. #2
    Eater of Worlds
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    One thing nice about cylinders is that the rate of change of height remains the same because of its straight sides.

    Cone volume = \frac{\pi}{3}r^{2}h

    But we are told 2r=h, so sub:

    V=\frac{\pi}{3}(\frac{h}{2})^{2}h

    V=\frac{{\pi}h^{3}}{12}

    Differentiate w.r.t h:

    \frac{dV}{dt}=\frac{{\pi}h^{2}}{4}\cdot\frac{dh}{d  t}

    We are told dh/dt=-80 and h=10.

    \frac{{\pi}(10)^{2}}{4}\cdot (-80)=-2000{\pi}

    Therefore, the volume of the cylinder is filling up at a rate of 2000{\pi} cubic cm per min.

    Since the cylinder has a 20 cm radius, it's volume is V={\pi}r^{2}h

    2000{\pi}=400{\pi}h

    h=5 \;\ cm/min
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  3. #3
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    Quote Originally Posted by CalcGeek31 View Post
    A liquid is going through a conical filter whose top diameter is the same as its height. It pours into a right circular cylinder whose radius is 20cm. How fast is the liquid in the cylinder rising in cm/min if the liquid in the filter is 10 cm high and falling at a rate of 80cm/min?

    I am not even sure where to start this question, any tips/help would be appreciated.
    A good place to start might be with the equations.

    What is the formula for the volume V of a cone with height "H" and radius "R"? Noting that, in this case, H = R what is the formula that you should use?

    What is the formula for the volume v of a cylinder with radius "r" and height "h"? Noting that, in this case, r = 20, what is the formula that you should use?

    You are given that, for the cone, dH/dt = -80, and are asked to find the rate of change dh/dt when H = 10. Can you think of a way to relate the various bits of information?
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