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Math Help - related rates

  1. #1
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    related rates

    Coffee is poured at a uniform rate of 20 cm^3/s into a cup whose inside is shaped like a truncated cone. If the upper and lower radii of the cup are 4 cm and 2 cm and the height of the cup is 4 cm, how fast will the coffee level be rising when the coffee is halfway up? [ hint: Extend the cup downward to form a cone.]

    i know that v = (1/3)(r^2)(h)
    i dont know how to set r as a function of h or vice versa.
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  2. #2
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    Quote Originally Posted by larryboi7 View Post
    Coffee is poured at a uniform rate of 20 cm^3/s into a cup whose inside is shaped like a truncated cone. If the upper and lower radii of the cup are 4 cm and 2 cm and the height of the cup is 4 cm, how fast will the coffee level be rising when the coffee is halfway up? [ hint: Extend the cup downward to form a cone.]

    i know that v = (1/3)(r^2)(h)
    i dont know how to set r as a function of h or vice versa.
    following the hint ...

    top radius = 4 cm

    height of the extended cone = 8 cm

    \frac{r}{h} = \frac{4}{8} = \frac{1}{2}

    r = 2h

    now work with the cone, assuming that it is filled to a height h = 4 cm at t = 0 ... the remaining void is the truncated cone (a frustrum).

    to answer the question, note that h = 6 cm from the cone's bottom when the liquid is halfway up the frustrum.
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