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Partial Derivatives and The Chain Rule

  1. Sep 30, 2009 #1
    1. The problem statement, all variables and given/known data
    The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 7 m and w = h = 9 m, and l and w are increasing at a rate of 6 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing.
    (a) The Volume
    (b) The Surface Area
    (c) The Length of the Diagonal


    2. Relevant equations
    L2=l2+w2+h2



    3. The attempt at a solution
    I have already done part a and part b but I'm having trouble with c.
    I tried to work it out and got 42m/s but this is incorrect. Some help please?
     
  2. jcsd
  3. Sep 30, 2009 #2

    Dick

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    42m/s is way too high. Just by gut feeling. How did you get that?
     
  4. Sep 30, 2009 #3
    2l(dl/dt)+2w(dw/dt)+2h(dh/dt)
    for A I got 675m^3/s
    for B I got 312m^2/s
     
  5. Sep 30, 2009 #4

    Dick

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    If L^2=l^2+w^2+h^2 then 2L(dL/dt)=2l(dl/dt)+2w(dw/dt)+2h(dh/dt). You want to solve for the dL/dt part. Don't forget to divide by the L. I didn't check the other ones. Do I need to?
     
  6. Sep 30, 2009 #5
    No the other ones are right. I have already checked them. How do I get L to know what to divide by? Do I just plug in the original l w h to find L?
     
  7. Sep 30, 2009 #6

    Dick

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    Absolutely.
     
  8. Sep 30, 2009 #7
    Thanks a lot. I figured it was something simple like that.
     
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