Sphere in a cone (ball in a wine glass)

In summary, the conversation discusses a problem involving finding the radius of a sphere that displaces the most volume when placed in a cone with height H and angle A. One approach suggested is to reduce it to a two dimensional case using rotational symmetry and setting up equations for the cone and circumscribed circle. The conversation is later corrected to refer to the problem as a ball in a martini glass.
  • #1
rcgldr
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sphere in a cone (ball in a martini glass)

A problem a friend mentioned to me years ago.

You have a cone with height H and angle A. What is the radius R of a sphere that when placed in the cone, displaces the most volume?

One suggestion was to reduce this to a two dimensional case.
 
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  • #2
Yeah, rotational symmetry would allow reduction to the 2-d case; so, set it up: what's the equation of a cone of angle A? the equation of the circle of radius R circumscribed therein? do the integral, get it in terms of R, differentiate w.r.t. R and set equal to 0.
 
  • #3
Correction: ball in a martini glass.

Glad I could help.
 

FAQ: Sphere in a cone (ball in a wine glass)

1. What is the concept of a "sphere in a cone" or "ball in a wine glass"?

The concept of a "sphere in a cone" or "ball in a wine glass" refers to a geometric shape where a sphere (or ball) is placed inside a cone (or wine glass) in a way that the sphere touches the base and the sides of the cone. This is often used as a visual representation in mathematics and physics to demonstrate concepts such as volume and surface area.

2. How is the volume of a "sphere in a cone" calculated?

The volume of a "sphere in a cone" can be calculated using the formula V = (1/3)πr²h, where r is the radius of the sphere and h is the height of the cone. This formula takes into account both the volume of the sphere and the cone, with the added volume of the sphere being equal to one-third of the volume of the cone it is placed in.

3. What is the significance of a "sphere in a cone" in real-life applications?

The concept of a "sphere in a cone" has many real-life applications, such as in architecture and engineering. It can be used to design structures such as domes and arches, as well as to calculate the volume of various objects that have a similar shape, such as ice cream cones and traffic cones.

4. Can a "sphere in a cone" have different dimensions?

Yes, a "sphere in a cone" can have different dimensions. The size and dimensions of the sphere and cone can vary, as long as the sphere touches the base and the sides of the cone. This means that the ratio of the sphere's radius to the cone's height can vary, resulting in different volume and surface area calculations.

5. How does the "sphere in a cone" relate to other geometric shapes?

The "sphere in a cone" is related to other geometric shapes, such as cylinders and pyramids. Just like how a sphere can be placed inside a cone, it can also be placed inside a cylinder or a pyramid in a similar manner. The calculations for volume and surface area will differ depending on the shape it is placed in.

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