Results 1 to 13 of 13
Like Tree6Thanks
  • 1 Post By Reckoner
  • 1 Post By Reckoner
  • 1 Post By Reckoner
  • 1 Post By Reckoner
  • 1 Post By Reckoner
  • 1 Post By Reckoner

Math Help - Related rates and derivatives word problem. Need help clearing a few things up.

  1. #1
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Related rates and derivatives word problem. Need help clearing a few things up.

    Hey all, thanks in advance for any help you can offer.
    I have completed most of the problem here, I just need some help in the right direction! Most of this problem will be trig, but I posted this in the calculus forum because it is calculus level. I will go ahead and post the whole problem in the form of an image to make things easier, with my questions at the bottom.





    My questions are from 46.1.2 and on. I have determined that the radius at the top of the cup is twelve, and I used the concept of similar triangles to express r as h.

    Could somebody please look this over?
    :

    Is this correct so far?

    Now is where my questions come in. 46.1.3 is driving me crazy. How do I find dh/dt? Here is my work so far:



    Is what I found for dv/dt correct? If it is, how do I go about finding dh/dt?



    Thanks for any help you can offer, and sorry for the wall of text!

    -dan
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    My questions are from 46.1.2 and on. I have determined that the radius at the top of the cup is twelve, and I used the concept of similar triangles to express r as h.
    The radius of the top of the cup is not twelve. Why are you assuming that the angle is \frac\pi4?
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Because they didn't give me enough information. How do I get the angle?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    Because they didn't give me enough information. How do I get the angle?
    You don't really need the angle. You know that the volume of a right circular cone is V=\frac13\pi r^2h. You are told that the volume of the cup is V = 36\pi\ \mathrm{cm}^3 and the height is h = 12\ \mathrm{cm}. Can you solve for r?
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Whoops! You sir are absolutely right! I found that 3 = r. Is the rest of this right?

    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    I found that 3 = r. Is the rest of this right?
    Well, I don't think you calculated \arctan\frac14 correctly. But again, you really don't need to find the angle. As the problem states, we have similar triangles. So

    \frac r3 =\frac h{12}

    \Rightarrow  r=\frac h4

    This is our "relation equation," as the problem calls it.
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Ok, I think I am starting to get it. Thank you! How does this look?


    Now we get down to derivatives. How do I get dh/dt? I think this is correct for dv/dt, except for missing information:


    Thank you so much!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    Ok, I think I am starting to get it. Thank you! How does this look?
    Looks good. Now solve the equation for \frac{dh}{dt}. Note that we already know \frac{dV}{dt} from the problem statement, and we can determine what h is at the time when there is 8\pi\ \mathrm{cm}^3 of slushy left.
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Ok, I feel like I am really close. How do I get h? This doesn't seem right...
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    Ok, I feel like I am really close. How do I get h? This doesn't seem right...
    h^3 = \frac{48V}\pi

    \Rightarrow h^3 = \frac{48(8\pi)}\pi = 384

    \Rightarrow h = \sqrt[3]{384} = 4\sqrt[3]6

    And we also have \frac{dV}{dt} = -0.1.

    Now substitute:

    \frac{dh}{dt} = \frac{16}{\pi h^2}\left(\frac{dV}{dt}\right) = -\frac{1.6}{16\pi\sqrt[3]{36}}

    = -\frac1{10\pi\sqrt[3]{36}}
    Last edited by Reckoner; June 4th 2012 at 05:36 PM.
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    How does this look? The final answer doesn't really look right to me...
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor Reckoner's Avatar
    Joined
    May 2008
    From
    Baltimore, MD (USA)
    Posts
    1,024
    Thanks
    75
    Awards
    1

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    Quote Originally Posted by dan147 View Post
    How does this look? The final answer doesn't really look right to me...
    That's what I got. So at that particular moment, the depth of the slushy is decreasing by about 0.010 cm/s.
    Thanks from dan147
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Newbie
    Joined
    Jun 2012
    From
    United States
    Posts
    7

    Re: Related rates and derivatives word problem. Need help clearing a few things up.

    What a problem! Thank you so much for your help, I really appreciate it. I thanked each one of your posts.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. related rates word problem
    Posted in the Calculus Forum
    Replies: 7
    Last Post: April 10th 2013, 03:01 PM
  2. Related Rates Word Problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 27th 2011, 07:08 PM
  3. related rates word problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 28th 2010, 08:44 PM
  4. Related Rates word problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 12th 2008, 06:41 AM
  5. related rates word problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 5th 2007, 02:29 PM

Search Tags


/mathhelpforum @mathhelpforum