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Math Help - Sphere cap Height calculations

  1. #1
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    Sphere cap Height calculations

    Hi,

    I know the formula for finding the volume of a sphere cap if radius and height are given,But how do I find the height of the cap/dome if only radius and volume are given? ie....r=5.4 volume = 160m3 (air space under dome).

    Thanks....Mark
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  2. #2
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    Quote Originally Posted by manich44 View Post
    Hi,

    I know the formula for finding the volume of a sphere cap if radius and height are given,But how do I find the height of the cap/dome if only radius and volume are given? ie....r=5.4 volume = 160m3 (air space under dome).

    Thanks....Mark
    1. Volume of a sphere cap:

    V= \pi r h^2 - \dfrac \pi3 h^3

    EDIT: (r is the radius of the base of the sphere!) Sorry for the confusion!

    2. Plug in the values you know. You get a cubic equation which could be solved by using Cardano's formula. You get an approximate result much easier and faster if you use an iterative method, for instance Newton's method.

    3. I've got h \approx 3.46355
    Last edited by earboth; November 29th 2009 at 07:22 AM.
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  3. #3
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    Re sphere cap

    Hi,
    Thanks for the reply.

    I used... pi/6(63.48 + 7.84)2.8 = V 104.6m3 (104.5605811) for the answer im looking for.

    How then do I plug the new values known ie r 5.4m and vol 160m3 into that to get the new height,can you show me an example?

    Thanks.....Mark
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  4. #4
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    Quote Originally Posted by manich44 View Post
    Hi,
    Thanks for the reply.

    I used... pi/6(63.48 + 7.84)2.8 = V 104.6m3 (104.5605811) for the answer im looking for.

    How then do I plug the new values known ie r 5.4m and vol 160m3 into that to get the new height,can you show me an example?

    Thanks.....Mark
    If I understand you correctly you used:

    V=\frac16 \pi h(3r^2+h^2)=\frac12 \pi r^2 \cdot h + \frac16 \pi \cdot h^3

    Now plug in the known values:

    160 = \frac12 \pi (5.4)^2 \cdot h + \frac16 \pi \cdot h^3

    This is a cubic equation. I already told you how to solve it.

    I've got h \approx 4.62132...
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  5. #5
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    Re sphere cap

    Hi,

    Can someone have a look attachment and let me know where im going wrong,cant get to the height "h"

    Thanks.....Mark
    Attached Thumbnails Attached Thumbnails Sphere cap Height calculations-05-12-2009-17-39-53_0035.jpg  
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