Yet another Trig Question

[Iamadam]

I really have absolutely no idea how to do this. I'm completely stumped!

Here is the question:

A cone with the radius 12cm has a semi-vertical angle of 30 degrees. Find the radius of the largest sphere which can just fit inside the cone.

If anyone could help I'd be verrrry happy, and alot less stressed
Thanks,
-Adam

 

[Gene]

You don't say what course this is from. That would have helped.
If the center of the sphere is at (0,r) the radius is perpendicular to the side and equals r. If it is d down from the apex you have two 30° {1,2,sqrt(3)} triangles.
One is
{r,d,r*sqrt(3)} = {r,2r,r*sqrt(3)}
And the other is
{12,24,12*sqrt(3)}
d+r = 2r+r = 3r = 12*sqrt(3)
r=4*sqrt(3)

PS. Just noticed the title.
r=d*sin(30)
d=r/sin(30)

tan(30) =
1/sqrt(3) =
12/(d+r) =
12/((r/sin(30))+r) =
12/((r/.5)+r) =12/3r
4/r=1/sqrt(3)
r=4*sqrt(3)

 

[Iamadam]

Thanks for that
Sorry...i feel really dumb saying this, but I have no idea what you did there...
Where did u get sqrt(3) from?

Any why is this...:
r=d*sin(30)
d=r/sin(30)

Sorry, it's probably blatently obvoius what you did, and I'm probably missing something stupid, but I don't quite understand it.

But thanks for the help anyway! I'd love if you (or anyone else) could explain a bit more, but if not, thats ok!
Thanks alot,
-Adam

 

[Gene]

The sides of a 30-60-90 triangle are in the ratio 1:2:sqrt(3). That's the source. I can't draw the pretty pictures that are so popular but half of it looks sorta like

  A
  |\
  | \
  |  \
  |   \B
 d|   /\
  |  /  \
  | /r   \
  |/      \
  |O       \ 
  |         \
 r|          \
  |           \
 C|____________\D

Triangle ABC is one right triangle, ACD is a second. Angle CAD is 30°
Sin(30)=r/d
Tan(30) = CD/AC
The center of the sphere is at O.
OB and OC are radii.

 

[Iamadam]

Thanks, I get it now