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Math Help - Surface area

  1. #1
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    Surface area

    I was wondering, how would you calculate the surface area of a sphere with a "cylinder" (I say "cylinder" rather than cylinder because the cylinder would have curved ends rather than flat) cut into it. The example I would use is that if you had a perfectly spherical apple and then cored it, how would you calculate the surface area of the cored apple?

    Daniel
    Last edited by wilsnunn; March 27th 2010 at 12:53 PM. Reason: typo
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  2. #2
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    Do you know the diameter of the core and the radius of the sphere? That should be quite enough to decide:

    1) How much of the sphere to subtract because you cut it away, and

    2) How much of the core to add because you exposed it.
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  3. #3
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    I was looking for an algebraic formula using R1 radius of sphere and R2 radius of area cut out, so you can get R3=R1-R2, if that makes any sense, sorry if it doesn't.

    So if we were given R1 and R2.

    Thanks

    Daniel
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  4. #4
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    Well, it takes a little work for the subtraction of the spherical caps, but the addition of the cyllindrical core is a right triangle probem.

    Considering only the top half the sphere (you can double it later), the height of the cyllinder is h = \sqrt{r_{1}^{2} - r_{2}^{2}}. Thus, the lateral surface area of the top half of the cyllinder is simply: \pi\cdot r_{2}^{2} \cdot h

    Now, you do the spherical caps to subtract. Come on, I expect the ancient Greeks could do it.
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  5. #5
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    Would it be at all possible to break that down a bit for me? (sorry) I am only an Additional Maths studnet, i have not even started AS level yet!

    Daniel

    Sorry
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  6. #6
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    Well, I don't know what any of that means, but you'll need two things for the cyllinder.

    1) Pythagorean Theorem. Hypotenuse is r1. One leg is r2. You must solve for the other leg. The solution is as I have posted it: h = ??

    2) Formula for the lateral surface area of a right circular cyllinder. \pi r^{2}h

    That's it.
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  7. #7
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    Quote Originally Posted by TKHunny View Post
    Well, I don't know what any of that means, but you'll need two things for the cyllinder.
    Ah thank you, and sorry i didnt see the american flag there I understand everything to do with the cylinder and the Pythagoras, it is just the curved tops of the cylinder that i am not sure how to calculate, sorry if i have caused any confusion.

    Thanks

    Daniel
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  8. #8
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    The surface area of one cap appears to be 2\pi r_{1}\left[r_{1}\;-\;\sqrt{r_{1}^{2}-r_{2}^{2}}\right]. You should check that out and make sure it looks reasonable. I'm sure there is some lovely geometry way to discern this, but I haven't managed to think of it this evening.

    It is not that hard a problem with a little calculus. This may be part of the reason why I failed to motivate myself to find the geometry solution.
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  9. #9
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    X
    Last edited by danielsdad; March 28th 2010 at 03:43 AM. Reason: got it wrong
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  10. #10
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    Last edited by danielsdad; March 28th 2010 at 03:44 AM. Reason: sorry, wrong!
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