Results 1 to 4 of 4

Math Help - related rates and upside down cones!

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    13

    related rates and upside down cones!

    So, my textbook only covers implicit diff. as it relates to ellipses and circles. Needless to say, it all went over my head. Now I have a related rates question where I absolutely need to use implicit diff. Here goes:

    A water tank has the shape of an inverted cone with a base radius of 4 meteres and a height of 6 meteres. If water is being pumped into the tank at a rate of 3 m^3/min, find the rate at which the water level is rising when the tank is 4 meters deep. Specify the units of your answer.

    So I have two triangles, one with measurements given, and one with only height (because radius at 4m deep is currently unknown)

    I also have the formula V = 1/3 pi*r^2*h

    Now, to set up a proportion to find r?

    r 4
    -- = --
    h 6

    6r = 4h, in terms of r

    r = 4h/6, substitute the known h

    r = 8/3m

    If I did everything right, I now know the radius of the cone when height = 4m. However, I'm sure I need to implicitly differentiate the volume formula and I'm not sure how.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,928
    Thanks
    757
    you need to get a volume formula in terms of a single variable.

    \frac{r}{h} = \frac{2}{3}

    r = \frac{2h}{3}

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

    V = \frac{4\pi}{27} h^3

    take the time derivative of the above volume function ... sub in your given values for \frac{dV}{dt} and h ... calculate the value of \frac{dh}{dt}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by angrynapkin View Post
    So, my textbook only covers implicit diff. as it relates to ellipses and circles. Needless to say, it all went over my head. Now I have a related rates question where I absolutely need to use implicit diff. Here goes:

    A water tank has the shape of an inverted cone with a base radius of 4 meteres and a height of 6 meteres. If water is being pumped into the tank at a rate of 3 m^3/min, find the rate at which the water level is rising when the tank is 4 meters deep. Specify the units of your answer.

    So I have two triangles, one with measurements given, and one with only height (because radius at 4m deep is currently unknown)

    I also have the formula V = 1/3 pi*r^2*h

    Now, to set up a proportion to find r?

    r 4
    -- = --
    h 6

    6r = 4h, in terms of r

    r = 4h/6, substitute the known h

    r = 8/3m

    If I did everything right, I now know the radius of the cone when height = 4m. However, I'm sure I need to implicitly differentiate the volume formula and I'm not sure how.
    In the problem, we are told that \frac{\,dV}{\,dt}=3~\frac{m^3}{min}. You are correct in saying that a proportion is need to get radius in terms of height: it should be \frac{r}{h}=\frac{4}{6}\implies r=\frac{2}{3}h

    Since the volume of a cone is V=\tfrac{1}{3}\pi r^2 h, we now see that V=\tfrac{1}{3}\pi\left(\tfrac{2}{3}h\right)^2h\imp  lies V=\tfrac{4}{27}\pi h^3

    Now [implicitly] differentiate both sides with respect to t:

    V=\tfrac{4}{27}\pi h^3\implies\frac{\,dV}{\,dt}=\tfrac{4}{9}\pi h^2\frac{\,dh}{\,dt}

    We can now plug in all the known values and solve for \frac{\,dh}{\,dt}.

    Can you try to take it from here?

    --Chris
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2008
    Posts
    13
    I believe so.

    Plug in 3m^3/min and 4

    Right side evaluates to 64pi/9

    Move 64/9 to left side

    27/64

    move pi to left side

    27/64pi m^3/min.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Need help with related rates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 14th 2009, 02:22 PM
  2. More Related Rates :D
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 12th 2009, 03:02 AM
  3. Related Rates
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 10th 2009, 02:55 PM
  4. Rates and Related Rates!!
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 2nd 2008, 10:53 AM
  5. rates and related rates
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 29th 2007, 09:51 PM

Search Tags


/mathhelpforum @mathhelpforum