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If water is poured into a cup at a rate of 1 cubic centimeter per second, how fast is the dept of the water increasing when the water is 4 cm deep?

r = 6 cm
h = 9 cm

V = 1/3(pie) r^2 x h
V = (pie)r'h + 1/3 (pie)r^2 h'

6 / 9 = r / h

6h = 9r
r = 3 / 4.5 h

V = 1/3(pie) (3/4.5h)^2

My answer came out to 9 / 64(pie) cm/sec.

  • Calculus -

    what is the shape of the container?

  • Calculus -

    The shape is a cone. The radius is 6 cm and the height is 9 cm

  • Calculus -

    Your answer is correct if you write it as

    I am curious why you went from
    6h = 9r to
    r = 3/4.5

    why not just do
    r = 6h/9 = 2h/3
    instead of having decimals in your fraction.
    Either we use fractions or we use decimals, but we generally don't mix them together.

  • Calculus -

    I think you have a mistake. I was working on this problem at the same time.

    V = 1/3 pi r^2 h
    Only 'r' is squared. It appears that you squared 'h' too.
    V = 1/3(pie) (3/4.5h)^2

    Anyway, even with squaring h too, I don't come up with 9/64 pi.

  • Helper - Calculus -

    Here is what I did
    from above
    r = 2h/3

    V = (1/3)π(r^2)h
    = (1/3)π(4h^2/9)h
    = (4π/27) h^3

    dV/dt = (4π/9)h^2 dh/dt
    when dV/dt = 1 and h = 4

    1 = (4π/9)(16)dh/dt
    dh/dt = 9/(64π)

    (notice that Katie had r = 3h/4.5 which I found strange since that is simply 2h/3 )

  • Calculus -

    I used r = 2/3 instead of r = 2/3 h!

    Thanks so much REINY.

    I really appreciate you taking the time to explain this to me.

    I am teaching myself calculus, and sometimes the book just doesn't explain things well.

    The r = 3h/4.5 threw me at first, until I figured out what she did (I had 2h/3).

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