Results 1 to 14 of 14

Math Help - Help cylinder volume optimization problem

  1. #1
    Member
    Joined
    Nov 2010
    Posts
    102

    Help cylinder volume optimization problem

    For the hollow cylinder shown, assume that R and r are increasing at a rate of 2 m/s, and h is decreasing at a rate of 3 m/s. at what rate is the volume changing when R= 7m and r =4m and h = 5m ?

    please help me on the path to solving this problem. I'm sorry the picture isn't perfect.
    Attached Thumbnails Attached Thumbnails Help cylinder volume optimization problem-untitled.png  
    Last edited by wair; November 26th 2010 at 07:55 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2010
    Posts
    102
    hmm does anyone know how to get the picture to show up ? with out having to click on it ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,741
    Thanks
    481
    V = \pi(R^2 - r^2)h

    take the derivative of the above equation w/r to time, then sub in your given values to determine \frac{dV}{dt}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2010
    Posts
    102
    Thanks again skeeter. I really appreciate your help.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Nov 2010
    Posts
    102
    I still can't solve this problem.. When I take the derivative of the equation I get 2pieR - 2pier but my H's are gone ? how do I take the derivative with respect to time correctly ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,741
    Thanks
    481
    Quote Originally Posted by wair View Post
    I still can't solve this problem.. When I take the derivative of the equation I get 2pieR - 2pier but my H's are gone ? how do I take the derivative with respect to time correctly ?
    product rule ... R, r, and h are all variables.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Nov 2010
    Posts
    102
    So I can't multiply everything out then take the derivative ? Also could you tell me how you came up with the equation ? ohh and what about the rates at which r, R, and h change that are given in the problem, are those just extra information ?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,741
    Thanks
    481
    Quote Originally Posted by wair View Post
    So I can't multiply everything out then take the derivative ? Also could you tell me how you came up with the equation ? ohh and what about the rates at which r, R, and h change that are given in the problem, are those just extra information ?
    you can multiply it all out but you still will have to use the product rule.

    this solid is a large cylinder with a smaller cylindrical "hole" in it ... \pi R^2 h - \pi r^2 h = \pi(R^2 - r^2)h

    as I stated earlier, once you get the derivative w/r to time sub in your given values for R, r, h, and their respective rates of change to determine dV/dt.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Nov 2010
    Posts
    102
    Ok so once I take the derivative I get, 2hRpi + piR^2 - pir^2 + 2hrpi. but now if i substitute I have two values of R,r and h. I have values of rates and I have values of size. kind of like position and velocity. so how do i relate the two together ?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,741
    Thanks
    481
    Quote Originally Posted by wair View Post
    Ok so once I take the derivative I get, 2hRpi + piR^2 - pir^2 + 2hrpi. but now if i substitute I have two values of R,r and h. I have values of rates and I have values of size. kind of like position and velocity. so how do i relate the two together ?
    your derivative is incorrect. are you familiar with the technique of implicit differentiation in solving related rates problems? It is important that you clearly understand that R, r, and h are all implicit functions of time.

    note the following derivative for the first term on the right side of the equation V = \pi R^2h - \pi r^2 h

    \displaystyle \frac{d}{dt} (\pi R^2 h) = \pi R^2 \cdot \frac{dh}{dt} + 2\pi R h \cdot \frac{dR}{dt}
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Nov 2010
    Posts
    102
    Ok is dh/dt equivilant to writing h prime ? ok I understand that now, so when I solve for any related rates problem I use implicit differentiation. ok so now the rate values would be substituted with the R prime h prime and r prime variables, or the dR/dt, dh/dt and dr/dt correct ?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Member
    Joined
    Nov 2010
    Posts
    102
    I get -122.52 m^3/s for this problem. Could you please confirm if this is correct ?
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,741
    Thanks
    481
    Quote Originally Posted by wair View Post
    I get -122.52 m^3/s for this problem. Could you please confirm if this is correct ?
    correct.

    \displaystyle \frac{dV}{dt} = -39\pi \, \frac{m^3}{s}
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Member
    Joined
    Nov 2010
    Posts
    102
    Thanks alot for all your help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 18th 2011, 01:08 PM
  2. Replies: 1
    Last Post: November 16th 2011, 09:11 AM
  3. Volume Optimization Problem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: May 25th 2009, 01:53 PM
  4. Optimization of cylinder
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 6th 2009, 07:27 AM
  5. Optimization Max volume problem. HELP!!!!
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 30th 2008, 12:44 PM

Search Tags


/mathhelpforum @mathhelpforum