How Do You Calculate the Volume of Liquid in a Partially Filled Conical Glass?

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Discussion Overview

The discussion revolves around calculating the volume of liquid in a partially filled conical glass, specifically determining the volume as a proportion of the full volume and finding the ratio of the height of the liquid to the height of the glass when it is half full. The scope includes mathematical modeling and reasoning related to geometry and volume calculations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant expresses difficulty in solving the problem and seeks assistance from the forum.
  • Another participant requests clarification on how the ratio H:2h was derived.
  • A participant explains the volume formula for a cone and attempts to relate the volumes of the full and half-filled cones, leading to confusion regarding the relationship between height and radius.
  • Another participant suggests considering the relationship between the radius and height using similar triangles to aid in the calculation.

Areas of Agreement / Disagreement

The discussion shows that participants are exploring different aspects of the problem, but there is no consensus on the correct approach or solution. Confusion remains regarding the relationship between height and radius in the context of the conical shape.

Contextual Notes

Participants have not fully resolved the mathematical relationships involved, particularly concerning the changing radius as the height of the liquid changes. The discussion also lacks clarity on the assumptions made in the volume calculations.

skiing4free
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This is a problem that my lecturer gave us in class and it has been bugging me ever since. I have been unsuccesful in finding or calculating a proper solution so I am hoping PF will be able to help...

This is the Q:

Let H be the height of a conical glass which is filled to a height h. Find the volume of the liquid in the glass as a proportion of the volume if the glass is full. Find the ratio h/H for which the glass is half full.
To answer this question you must construct a mathematical model defining all the variables.

This does not sound like a difficult question to solve but whenever I try to solve with simple conical volume equations I just get that the ratio is H:2h which is obviously not right. Any help would be greatly appreciated.
 
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Why don't you explain how you got H:2h?
 
Well the formula for a volume of a cone is:
V=1/3*pi*r^2*H.

V1 which is the volume of the full cone and V2 is the volume of the half full cone it is clear that V1/2=V2. I put the volume equations into this with the two different heights and when the constants are removed (pi, 1/3) you are left with H=2h. ahh but i have just seen how the radius would of course change... Now I think I am more confused
 
Can you find a relationship between what the radius and the height is going to be when you're at a certain height up the cone? (hint: think similar triangles)
 

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