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Good ole hemispherical dome differentials problem

  1. Nov 1, 2008 #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 0.03 cm thick to a hemispherical dome with diameter 54 m. (Round the answer to two decimal places.)

    2. Relevant equations
    V = 1/2(4/3(pi*r^3)) = 1/2(4/3(pi*(1/2D)^3)) = 1/2(4/3*pi(1/8D^3)) = 1/12pi*D^3


    3. The attempt at a solution
    How does V change if we change d from 54m to (54 = 0.03*10^-2)m?
    dV/dD = 1/4*pi*D^2
    dV = 1/4*pi*D^2*dD
    = 1/4*pi*(54)^2(0.03*10^-2)

    I got 0.679 as an awesome and got it wrong...
     
  2. jcsd
  3. Nov 1, 2008 #2

    HallsofIvy

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    Why did you switch to diameter? Since you are only painting a hemisphere, the paint only increases the radius, not the diameter. And use [itex]V= (2/3)\pi r^3[/itex].
     
  4. Nov 1, 2008 #3
    Okay, using 2/3 seems a lot easier, but my professor gave an example in class of converting radius to diameter. Furthermore, I had a problem on a previous homework where I had to solve for the rate at which a sphere increased, and I got the right answer after converting the radius in the formula into diameter after attempting to solve for the rate at which the radius changed. If I do go about solving for the radius in this problem, what all needs to be done to address the problem of solving for diameter?
     
  5. Nov 1, 2008 #4

    Dick

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    You can use the diameter if you want. But after applying the paint the diameter becomes 54m+2*(0.03cm).
     
  6. Nov 2, 2008 #5
    okay. thanks for the help, all!
     
  7. Mar 11, 2011 #6
    Your original differential seems to be correct, you used sound mathematical principles, the only problem I see is the final answer. When I put .25*pi*(54)^2*.0003 I get .687. I have tried to figure out what you might have miss entered, but the formula you used was good.
     
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