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Thread: Rate of Change: Volume of Water in a Cone

  1. #1
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    Rate of Change: Volume of Water in a Cone

    A cone-shaped coffee filter of radius 6cm and depth 10cm contains water, which drips out through a hole at the bottom at a constant rate of 1.5cm^3 per second.

    a) If the filter starts out full, how long does it take to empty?


    v = (1/3)(pi)(r^2)h - 1.5t
    0 = (1/3)(pi)(6^2)(10) - 1.5t
    0 = 376.99 - 1.5t
    376.99 = 1.5t
    t = 251.33 sec

    b) Find the volume of water in the filter when the depth of the water is h cm.

    v = (1/3)(pi)(r^2)h - 1.5t
    v + 1.5t = (1/3)(pi)(r^2)h
    h = (v + 1.5t) / ((1/3)(pi)r^2)

    c) How fast is the water level falling when the depth is 8cm?

    8 = (v + 1.5t) / ((1/3)(pi)r^2)

    How do I solve this with so many unknown variables (volume, time and radius)? My suspicion is that I did part (b) incorrectly. Can anyone please show me where I went wrong? Thanks!

  2. #2
    Senior Member
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    Re: Rate of Change: Volume of Water in a Cone

    A cone-shaped coffee filter of radius 6cm and depth 10cm contains water, which drips out through a hole at the bottom at a constant rate of 1.5cm^3 per second.

    a) If the filter starts out full, how long does it take to empty?

    v = (1/3)(pi)(r^2)h - 1.5t
    0 = (1/3)(pi)(6^2)(10) - 1.5t
    0 = 376.99 - 1.5t
    376.99 = 1.5t
    t = 251.33 sec

    b) Find the volume of water in the filter when the depth of the water is h cm.

    v = (1/3)(pi)(r^2)h - 1.5t
    v + 1.5t = (1/3)(pi)(r^2)h
    h = (v + 1.5t) / ((1/3)(pi)r^2)

    c) How fast is the water level falling when the depth is 8cm?

    8 = (v + 1.5t) / ((1/3)(pi)r^2)
    Your answer to part b) is pretty simple: v = (1/3)(pi)(r^2)h. However, we do have information that ties r and h together. The ratio is

    r/h = 6/10 = 3/5

    Or in other words,

    r = 3h/5

    So we can rewrite v = (1/3)(pi)(r^2)h as

    v = (1/3)(pi)((3h/5)^2)h = [3pi(h)^2]/25

    When we are asked, “How fast is the water level falling?”, it means, “What is dh/dt?”

    v = [3pi(h)^2]/25

    dv/dt = (6pi(h)/25)(dh/dt)

    (dh/dt) = (dv/dt)/ (6pi(h)/25)

    Just plug in dv/dt and h for your answer.

    Hope that helps.

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