Maximizing Ice Cream Cone Volume: Solving the 30° Cone Problem

  • Thread starter jonnhannah
  • Start date
  • Tags
    Cone Ice
In summary: For the ball, it's the maximum volume. For the part of the ball in the cone, I think it's the maximum volume of the part of the ball in the cone.
  • #1
jonnhannah
6
0
:confused:
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!?

PLEASE HELP!?

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 doesn't 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:
Physics news on Phys.org
  • #2
jonnhannah said:
:confused:
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!?

PLEASE HELP!?

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 doesn't 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...

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):
Maths.jpg
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 [tex]V = \frac{4}{3} \pi R ^ 3[/tex], 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
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
oedipa maas said:
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.

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
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
jonnhannah said:
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.

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
for the cone i believe the imaginary R = .2679 for circular plane at top, but the R for sphere is a variable still...
 
  • #8
jonnhannah said:
for the cone i believe the imaginary R = .2679 for circular plane at top, but the R for sphere is a variable still...

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
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
jonnhannah said:
:confused:
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!?

Wait, just want to 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
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
r for cone i believe 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?
 

1. What is the "Ice Cream Cone Problem"?

The "Ice Cream Cone Problem" is a mathematical problem that involves determining the maximum amount of ice cream that can be contained in a cone-shaped ice cream cone without any of the ice cream spilling out.

2. What makes the "Ice Cream Cone Problem" challenging?

The "Ice Cream Cone Problem" is challenging because it requires the use of mathematical concepts such as geometry, calculus, and optimization to find the optimal solution.

3. Is there a specific formula for solving the "Ice Cream Cone Problem"?

Yes, there is a formula for solving the "Ice Cream Cone Problem" known as the "Cone Volume Formula." It is V = 1/3πr²h, where V is the volume, r is the radius of the cone's base, and h is the height of the cone.

4. Can the "Ice Cream Cone Problem" be solved in real-life situations?

Yes, the "Ice Cream Cone Problem" is applicable in real-life situations, such as determining the optimal size of an ice cream cone to minimize waste or finding the maximum amount of ice cream that can be served in a cone without any spillage.

5. Are there any practical applications of solving the "Ice Cream Cone Problem"?

Yes, there are practical applications of solving the "Ice Cream Cone Problem" in various fields such as engineering, architecture, and food industry. It can also be used to optimize the design of ice cream cones for maximum efficiency.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
17
Views
4K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
28
Views
16K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
  • Calculus and Beyond Homework Help
Replies
7
Views
11K
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
5K
Replies
5
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
Back
Top