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Math Help - calculus volume problem

  1. #1
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    calculus volume problem

    From Calculus 1:

    A ball of radius 13 has a round hole of radius 7 drilled through its center. Find the volume of the resulting solid.

    Okay, I have no idea what I am doing wrong here:

    General plan is to find the volume of the ball and then subtract the volume of the part drilled out.

    The volume of the ball:
    V = 4/3pi(13^3) = 9202.77208

    volume of the part drilled:
    V = Hpi(7^2) = 26 * pi * 49 = 4002.389041

    So I get a final answer of about :
    5200.38

    But it's the wrong answer. Help??
    Last edited by littlejodo; December 1st 2008 at 08:29 PM.
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  2. #2
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    Well I would recomend coming up with some system of equations and geting to a single variable. From there try performing some calculus. Also consider that you are not actually removing a right cylinder from the sphere - it does not have flat ends. I hope that helps get you going.
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  3. #3
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    volume

    Take the sphere as its centre at the origin. The volume you found for the sphere is correct. The area of the cylinder in the middle needs to be subtracted. Because the hole drilled has a radius of 7, this line is x=7.

    Rotating the line x=7 about the y axis gives V = \pi \int \limits_{-6.5}^{6.5} 7^2  dy

    Integrating you get \pi [49y] with limits -6.5 and 6.5
    \pi * [(49 * 6.5) - (49 * -6.5)] = 13 * 49 \pi = 2001.19452<br />
    Now subtract this from the sphere volume and you should end up with the correct answer.
    Last edited by nzmathman; December 1st 2008 at 09:00 PM. Reason: made mistake, changed dx to dy
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  4. #4
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    Why did you use 13 as the height for the cylinder and not 26? I thought if the radius of a sphere is 13 then the height would be the same as the diameter.

    why is my sphere volume wrong? I thought the formula for it is
    V = pi r^3

    The only thing I can think of is to find it with integration...

    something like...

    pi r^2 integrated from 0 to 13?

    pi [r^3/3] with 13 as r ... V = 2300.69302

    but that doesn't give me the right answer either... I'm confused. help?
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  5. #5
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    volume

    littlejodo, sorry about that mistake I made.

    My approach would be:

    The sphere should be 4/3 * \pi * 13^3
    The cylinder should be \pi * 7^2 * 26

    Subtract the cylinder volume from the sphere volume which gives 5200.38. Which is apparently wrong? Could you post the correct answer please?
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  6. #6
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    I don't have the correct answer. I do submissions online and it just tells me when I'm wrong.

    I think you were on to something when you used the integration, I am just not sure how to accomplish that for the sphere as well.
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  7. #7
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    Does anybody think this could be set up as a solid of revolution problem? Would I rotate around the x or y axis, if so?

    Thanks!
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