anonymous
  • anonymous
Can anyone help me to determine the spherical cap volume with only 2 variables available, which are the sphere radius and the sphere cap diameter
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
What are the variables?
anonymous
  • anonymous
spherical cap diameter and sphere radius
baru
  • baru
do you have a diagram?

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IrishBoy123
  • IrishBoy123
you could do it as a volume integral if you have diameter D and radius R, you can say that \(\sin \phi = \dfrac{D}{2R}\) where \(\phi\) is the polar angle. so the volume of the cone, ie less the cap, is: [\( \star\)] \(\large 2\pi \; \int\limits_{r = 0}^{R} \; \int\limits_{\phi = 0}^{\arcsin (\frac{D}{2R})}\; r^2 \sin\phi \,dr \,\, d\phi .\) LESS [\(\square\)] the volume of the cone from the usual formula... i can't get wolfram to do \(\star\) for me but i think it should be: \(2 \pi \dfrac{R^3}{3} \left(1-\sqrt{\dfrac{1-D^2}{4 R^2}}\right)\) as it is doing the inner integral as follows: http://www.wolframalpha.com/input/?i=+int_%7Bp+%3D+0%7D%5E%7Barcsin%28D%2F%282R%29%29+%7D++++sin%28p%29++++dp+ and \(\square\) is \(\dfrac{1}{3}\pi \left(\dfrac{D}{2} \right)^2 \sqrt{R^2 - \left(\dfrac{D}{2}\right)^2}\) or summat like that :-)
wolf1728
  • wolf1728
It seems to me the problem is calculating the spherical cap height when just the sphere radius and the spherical cap radius are known. I have made a graphic which is attached. We can easily calculate the small green line (spherical cap radius minus radius) but can we determine the spherical cap HEIGHT from this information?
1 Attachment
wolf1728
  • wolf1728
Heck, the cap height can easily be solved from cap radius and sphere radius. height2^2 = radius^2 - cap radius ^2 Then finding the cap height is just a matter of subtracting height 2 from the radius.
anonymous
  • anonymous
@wolf1728, indeed i can calculate the spherical cap height by subtracting sphere radius to square root of R^2 - r^2 (based on phytagorean theorem
wolf1728
  • wolf1728
Yes, it occurred to me too and I just came back to post that! LOL
wolf1728
  • wolf1728
And that seems a LOT easier than calculating a volume integral.
anonymous
  • anonymous
thank youso much, wolf ..... :)
wolf1728
  • wolf1728
U R welcome!
anonymous
  • anonymous
now i think i can close this
wolf1728
  • wolf1728
Okay!
anonymous
  • anonymous
nice meeting u
IrishBoy123
  • IrishBoy123
you were asked to calculate the volume of the spherical cap
IrishBoy123
  • IrishBoy123
https://en.wikipedia.org/wiki/Spherical_cap will help if you just want to use a formula....
wolf1728
  • wolf1728
But the formula they show is for spherical cap volume IF you know the cap height and radius. We were determining the formula if you only know the spherical cap radius and the sphere radius.
IrishBoy123
  • IrishBoy123
yep, so now plug the height in to get what they want, ie the volume :p

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