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Calculus Help: Differentials (Rates)

  1. Oct 22, 2004 #1
    A hemispherical dome with radius 48 feet will be given a coat of paint 0.07 inches thick. Use differentials to estimate the number of gallons of paint that will be needed to paint the dome.

    Here's what I did:
    f(x+h) = f '(x)h + f(x) and v=4/3pi*r^3; r=48ft.

    volume of the dome:
    V=4/3pi*r^3 = f(r)

    I set (r+h) to be the radius of the paint volume
    so, V=4/3pi*(r+h)^3 = f(r+h)

    then, f(r+h) = f '(r)h + f(r)

    The question just want the volume of paint needed, so I subtracted f(r)
    f(r+h)=f '(r)h + f(r) - f(r), simplify

    then, f(r+h)=f'(r)h

    derivative, V(paint) = 4/3pi*r^3, then dv/dr(paint)=4pi*r^2

    given: r=48 feet, h=0.07inches = 0.0058333feet

    f(r+h)=f'(r)h => f(r+h) = 4pi*(48)^2*0.0058333

    I got approximately 168.891056 gallons, but the correct answer is 631.7000268 gallons.

    What did I go wrong?
     
  2. jcsd
  3. Oct 22, 2004 #2

    Hurkyl

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    Have you checked all of your work? (even the problem and your starting points)


    Oh, and BTW,

    You know full well what you have written is wrong: the left hand side of this last equation should not be f(r+h).
     
  4. Oct 22, 2004 #3
    I just realized the question asks for a hemispherical dome, not a sphere. Can someone please help me with this question? Thanks
     
  5. Oct 22, 2004 #4

    Hurkyl

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    Doesn't this realization suggest a change to your attempt at solving it?
     
  6. Oct 22, 2004 #5
    I keep getting the wrong answer. :confused:
     
  7. Oct 23, 2004 #6

    Hurkyl

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    Have you checked your arithmetic and conversions as well? For instance, I entered this into google:

    4 * pi * (48 feet)^2 * (0.07 inches) in gallons

    and the result was 1,263 US gallons, not then 168 you got.
     
  8. Oct 23, 2004 #7
    Find the volume of the outer sphere and the inner sphere. What is the volume between the spheres? How can differentials be used to get a volume? Also, the choice of a good (bad) coordinate system will make the problem much easier (harder).
    /s
     
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