Partial Derivatives and The Chain Rule

  • Thread starter ktobrien
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  • #1
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Homework Statement


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


Homework Equations


L2=l2+w2+h2



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?
 

Answers and Replies

  • #2
Dick
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42m/s is way too high. Just by gut feeling. How did you get that?
 
  • #3
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2l(dl/dt)+2w(dw/dt)+2h(dh/dt)
for A I got 675m^3/s
for B I got 312m^2/s
 
  • #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?
 
  • #5
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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?
 
  • #6
Dick
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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?
Absolutely.
 
  • #7
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Thanks a lot. I figured it was something simple like that.
 

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