# Ice Cream Cone Problem

1. May 23, 2007

### jonnhannah

PROBLEM: A cone with a 30degree angle and a hieght of 1 must fit a sphere of icecream in it with a maximum volume.
what is that volume, and what percentage of the sphere is in that cone!???

this is all i have
R= (h-a)(sin15)
a=distance between center of sphere and imaginary plane on top of cone.
i think R=0.2679 for the cone, but that doesnt help much, just basic law of sines... i'm not sure how to set up the dirrivative to maximize V for sphere with this information...

Last edited: May 23, 2007
2. May 23, 2007

### VietDao29

Err... I don't think we need any derivatives to work out the problem.
The center of the spherical ice-cream ball should always be on the Ox axis, so that it can fit the cone (see attachment):
The radius of the sphere is the distance between the center and one of the two sides of the cone.
Note that the volume of the sphere is $$V = \frac{4}{3} \pi R ^ 3$$, so when R is largest, V will also be largest, right?
Looking at the image, what's the maximum value of R?
Can you go from here? :)

3. May 23, 2007

### oedipa maas

Can't you balance an infinitely large sphere of ice cream on top of the cone?

If the radius of the sphere is the same as the radius of the opening of the cone, the ice cream ball will just rest on the lip of the cone. But if the ice cream ball is much bigger than the cone, the ball will still rest on the lip, but a much smaller portion of the ice cream will protrude into the cone.

4. May 23, 2007

### VietDao29

Nope, then that ice-cream ballie would not fit the cone. It's just how we define "fit".
You can, of course put an infinitely small hat on, and balance it, but, surely, it does not fit your head.

5. May 23, 2007

### jonnhannah

true, but it must fit inside the cone! this is the problem i'm having. i need the volume INSIDE the cone of the one scoop of icecream... otherwise, yes, i would have an easy answer of infinite units cubed.

6. May 23, 2007

### VietDao29

Have you found out the maximum value for R from my previous post? It should be easy, since you have the height of the cone, and the angle.

7. May 23, 2007

### jonnhannah

for the cone i believe the imaginary R = .2679 for circular plane at top, but the R for sphere is a variable still...

8. May 23, 2007

### VietDao29

Ok, look at the attachment again, I've edited it. A little bit messy, but hope you can see it. :)
So, we have:
OH = 1 (problem stated).
Angle AOH = 15 degrees.
C is the center of the sphere.
We also know that AC is perpendicular to OA (so that an ice-cream should fit in the cone). The radius R of the sphere is AC. Can you find AC?

9. May 23, 2007

### Werg22

While I won't say that the case in which the sphere does not fall tangentially in the cone is the case for which the submerged volume is maximum, I won't venture to say that it is, the other cases have to be verified also to be fully rigorous.

10. May 24, 2007

### VietDao29

Wait, just wanna ask, is this the maximum volume for the ice-cream ball, or the maximum volume of the part of that ice-cream ball in the cone?

11. May 29, 2007

### jonnhannah

EXACT PROBLEM:
you place a sphere of ice cream into a cone of hieght 1
1. what radius of the sphere will give you the most volum of ice cream inside the cone (as opposed to above the cone) for a cone of base 30 degrees?
2. what is that volume inside the cone and what percentage is inside the cone and outside the cone?

12. May 29, 2007

### jonnhannah

r for cone i beleive is .2679
and i'm not sure how to solve for r of sphere to achieve a maximum volume inside the cone...
i did calculate one theory that i had and found radius to be .230
any thoughts?