Largest possible volume of a cylinder inscribed in a cone

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Homework Help Overview

The problem involves finding the largest possible volume of a right circular cylinder that is inscribed in a cone with a specified height and base radius. The original poster expresses confusion about how to utilize the volume formulas for both the cone and the cylinder in this context.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants suggest drawing a coordinate system to visualize the cone and the inscribed cylinder, questioning the equations of the lines that represent the cone's sides. There is also a focus on identifying the dimensions of the cylinder based on the cone's geometry.

Discussion Status

The discussion is ongoing, with participants exploring the geometric relationships involved. Some guidance has been provided regarding the importance of visual representation, but no consensus or definitive approach has emerged yet.

Contextual Notes

There is a noted lack of specific dimensions or equations for the cylinder or cone, which may affect the ability to derive a solution. Participants are encouraged to clarify these aspects as part of their exploration.

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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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?
 


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


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.
 

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