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Rate of change

  1. Sep 16, 2009 #1
    1. The problem statement, all variables and given/known data
    Use the formula V = (1/3)pi r2(h) for the volume of a right circular cylinder to find
    a. the average rate at which the volume of a right circular cylinder changes with the radius r as r increases from 3 cm to 6 cm.
    b. the instantaneous rate at which the volume of the right circular cylinder changes with r when r = 6 cm.

    I tried to solve it at home and I arrived at the answers:

    a. 3(pi)(h)

    b. 4(pi)(h)

    Are my answers correct?

    thanks
     
  2. jcsd
  3. Sep 16, 2009 #2

    Mark44

    Staff: Mentor

    Without working the problem for myself, I don't know. Show us how you got these answers. In this problem is only the radius changing, or are both radius and height changing?
     
  4. Sep 16, 2009 #3

    LCKurtz

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    Gold Member

    That isn't the formula for the volume of a cylinder. It is the formula for the volume of a cone.
     
  5. Sep 16, 2009 #4
    Solution:

    dy/dx=( 1/3 π(〖6)〗^(2 ) h- 1/3 π(〖3)〗^2 h)/(6-3)

    dy/dx=( 1/3 π36h- 1/3 π9h)/2

    dy/dx=( 1/3 π36h- 1/3 π9h)/2
    dy/dx=(12πh-3πh)/3
    dy/dx=9πh/3
    =3πh
    Solution:
    dy/dx=2/3 πrh
    At r = 6:
    dy/dx=2/3 π6h
    =4πh
     
  6. Sep 16, 2009 #5

    LCKurtz

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    Your formula for volume is V = (1/3)pi r2(h). There is no y or x in this equation so dy/dx makes no sense. You are talking about V as a funtion of r and apparently h is a constant. To get the average rate of change as r changes from 3 to 6 you want to calculate:

    [tex]V_{ave} = \frac {V(6)-V(3)}{(6-3)}[/tex]

    That is what you actually started to calculate, but you mis-labeled, it isn't a derivative, and you need to check your arithmetic.

    For part (b) you want dV/dr. Calculate it, but don't put the numbers in for r until your have the derivative simplified. I think you should get 24 pi h. No y or x anywhere in your answer.
     
  7. Sep 16, 2009 #6

    Mark44

    Staff: Mentor

    What are those things around the 6 and the 3?
     
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