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Differentials and paint needed problem

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50m

    2. Relevant equations

    A= 2(pi)r2

    3. The attempt at a solution

    dr=0.0005 m
    r=25m
    dA=?

    (A=2(pi)r2)'

    dA= 4(pi)r*dr

    I wont go any further because the number is very small and i think incorrect.
    I am pretty sure I'm missing a step but I can't figure out what or why.
    Then again I could be way off.
     
  2. jcsd
  3. Oct 28, 2009 #2

    lanedance

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    Re: differentials

    the amount of paint will be a volume, not area, think of a thin spherical shell
     
  4. Oct 28, 2009 #3

    HallsofIvy

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    Re: differentials

    What is the formula for volume of a sphere?

    ([itex]\pi r^2[/itex] is the area of circle and not relevant here.)
     
  5. Oct 28, 2009 #4
    Re: differentials

    volume of a sphere= (4/3)pi*r^2

    volume of hemisphere= (4/6)pi*r^2

    dv= (4/3)pi*r*dr

    dv= (4/3)*pi*25*.0005

    am I wrong still?
     
  6. Oct 28, 2009 #5

    lanedance

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    Re: differentials

    yes...

    volume of a sphere is (4/3)pi*r^3

    and when you differntiate a power, you multiply by the orgiginal power, not divide
     
  7. Oct 28, 2009 #6

    lanedance

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    Re: differentials

    that looks like the whole sphere, how about the hemisphere part?
     
  8. Oct 28, 2009 #7
    Re: differentials

    dv=2pi*r^2
     
  9. Oct 28, 2009 #8

    lanedance

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    Re: differentials

    almost.... you just need to add a dr in there

    so to sumamrise
    V(r) = (1/2)(4/3)pi.r^3 = (2/3)pi.r^3

    then the derivative is
    dV/dr = 2pi*r^2

    so for a small change in r, ∆r the approximate corresponding change in ∆V volume will be
    ∆V= (dV/dr).∆r = 2pi.r^2.∆r
     
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