Calculus

a pond that lies in a crater is being drained at the rate of 1000m3/min.If the crater has the shape of an inverted cone of radius 200(at he original water level) and original depth is 50 m (at the center) find that rate at which the water level begins to fall.How fast is it falling when the water at the center is only 20 m deep?

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  1. The important thing to realize here is that the rate of water loss is the surface area times the rate of depth change.
    dV/dt = pi r^2 dh/dt
    where h is the depth and both dV/dt and dh/dt are negative

    -1000 m^3/min = pi (200^2)dh/dt
    when h = 50

    then what is r when h = 20?
    r is proportional to h (cone geometry
    so
    20/50 = r/200
    r = 400/5 = 80
    so then
    -1000 = pi(80^2) dh/dt

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  2. Thaanks that helped a lot <3

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  3. Shouldnt you have used the formula of volume of cone which is 1/3 pi r^2 h . And the answer should be 8 or 50 but by your procedure i m not getting it .I appreciate you help.

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  4. No, you do not need volume of cone, just the surface area of the water.
    Look more formally
    V = (1/3) pi r^2 h
    where r = (200/50) h = 4 h
    V = (1/3) pi (16) h^3
    dV/dh = 16 pi h^2
    dV = 16 pi h^2 dh
    dV/dt = 16 pi h^2 dh/dt
    but h = r/4
    so
    h^2 = r/16
    so
    dV/dt = pi r^2 dh/dt
    which you could see immediately by considering a slice of depth dh at the surface.

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  5. Okaay thaat makes sense but do i get 8 or 50 as your answer if u solve it fully .thanks .

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  6. -1000 m^3/min = pi (200^2)dh/dt
    when h = 50
    -----------------------
    so
    dh/dt = -.008 meters/minute = - 0.8 cm/min
    -------------------

    then what is r when h = 20?
    r is proportional to h (cone geometry
    so
    20/50 = r/200
    r = 400/5 = 80
    so then
    -1000 = pi(80^2) dh/dt
    ------------------
    so
    dh/dt = - .05 m/min = - 5 cm/min

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  7. Thanks

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