Largest possible volume of a cylinder inscribed in a cone

In summary, the problem is to find the largest possible volume of a right circular cylinder inscribed in a cone with height h and base radius r. The equations for the volume of a cone and a cylinder are given, but the dimensions and equations for the cylinder and cone are not stated. A suggestion is given to draw a picture and label important points.
  • #1
Calculus!
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Homework Statement



A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

I'm just really confused on how to figure this one out. The equation for the volume of a cone is v = 1/3pi r^2h and the volume of a cylinder is v = pi r^2h. I just don't know how to use these two formuals in order to find a solution. Please help me. Thanks.
 
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  • #2


Draw a picture. Draw a coordinate system so and two lines, one through (r, 0) and (0, h) and the other through (-r, 0) and (0, h). That represents your cone. What are the equations of the lines? A cylinder inside the cone is represented by a rectangle in your picture. What are the dimensions of that cylinder?
 
  • #3


The problem didn't state any dimensions or equations for the cylinder or cone.
 
  • #4


Calculus! said:
The problem didn't state any dimensions or equations for the cylinder or cone.

From your first post:
A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

r and h are the dimensions of the cone. Follow Halls's suggestion to draw a picture and label the important points.
 

FAQ: Largest possible volume of a cylinder inscribed in a cone

1. How do you calculate the largest possible volume of a cylinder inscribed in a cone?

To calculate the largest possible volume of a cylinder inscribed in a cone, you will need to use the formula V = 1/3 * π * r^2 * h, where r is the radius of the cone's base and h is the height of the cone. This formula will give you the maximum volume of the cylinder that can fit inside the cone.

2. What is the relationship between the cone and cylinder in this scenario?

The cone and cylinder are both 3-dimensional geometric shapes. In this scenario, the cylinder is inscribed, or nested, inside the cone. This means that the base of the cylinder touches the base of the cone and the top of the cylinder touches the tip of the cone.

3. Can the volume of the cylinder be larger than the volume of the cone?

No, the volume of the cylinder cannot be larger than the volume of the cone in this scenario. This is because the cylinder is inscribed inside the cone, meaning it must fit completely within the cone's dimensions. Therefore, the volume of the cylinder must always be smaller than the volume of the cone.

4. How does the height of the cone impact the volume of the cylinder?

The height of the cone directly affects the volume of the cylinder. As the height of the cone increases, the maximum possible volume of the cylinder also increases. This is because a taller cone allows for a taller and wider cylinder to fit inside, resulting in a larger volume.

5. Is there a real-life application for calculating the largest possible volume of a cylinder inscribed in a cone?

Yes, there are several real-life applications for this calculation. For example, in engineering and architecture, this concept can be used to determine the maximum volume of a cylindrical tank that can be placed inside a conical storage structure. It can also be used in packaging design to determine the maximum amount of liquid or solid product that can fit inside a conical container.

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