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Homework Help: Rearranging to eliminate the R variable in volume equation

  1. Nov 10, 2011 #1
    A water glass (10 cm diameter at the top, 6 cm diameter at the bottom, 20 cm in height) is being filled at a rate of 50 cm^3/min. Find the rate of change of the height of the water after 5 seconds.

    V=(∏* (h/3)) * (R^2 + r^2 +(R*r))

    I've been messing around with this for a bit and made a bit of leeway but I'm unsure of how to eliminate R and r from my equation. If it was a cone I would use the r/h=5/20 and rearrange to 5h/20 and replace my 'r' then differentiate. However that ratio doesn't seem to work on a truncated cylinder and I have two radius-es. Any suggestions?
     
  2. jcsd
  3. Nov 10, 2011 #2
    What are R, r, and h?
     
  4. Nov 10, 2011 #3
    R=5 cm r= 3 cm h= 20 cm
     
  5. Nov 10, 2011 #4

    Dick

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    Ok. So r is your lower radius. That's a constant. As the glass fills R and h will change. But there is a linear relation between them since the edge of the glass is a straight line. When h=0 cm then R=3 cm. When h=20 cm R=5 cm. What's the linear relation?
     
  6. Nov 11, 2011 #5
    OK cool I think I get what you're going at. If I put those point on a graph the slope would be a slope of R/H=10/1. I can sub that in for R and put 3 in for r because it's a constant. Simplified it might look a little something like this

    V= 100H^3(1/3)∏ + 90H^2(1/3)∏ + 9H(1/3)∏

    I'm still a little uncertain what to do with the issue of time. Could I take the derivative of the height (20 cm) ,when the glass is full, get the rate of change and then attempt to find the rate algebraically?
     
  7. Nov 11, 2011 #6

    Dick

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    But R=10h doesn't work. Remember, when h=0 cm then R=3 cm. When h=20 cm R=5 cm. If I put h=0 into R=10h I get 0, not 3. Can you fix that? For the other part of your question how much water is in the glass after 5s? Use that to find R and h at the time you want to find the rates.
     
  8. Nov 11, 2011 #7
    Could I use R=5H/20 and plug 3 in as a constant? If you plug in 0, the answer will still be 0 however I am a little unsure what else to try since I cant think of a number that i can multiply times 0 and not get 0, unless I just add 3 on to the end like

    R= 5h/20 + 3
     
  9. Nov 11, 2011 #8

    Dick

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    R=5h/20+3 is ok, for h=0. But at h=20 you want R=5, it's NOT ok there. Fix the slope. What should be multiplying h instead of 5/20?
     
    Last edited: Nov 11, 2011
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