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

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SUMMARY

The discussion focuses on calculating the volume of liquid in a partially filled conical glass, specifically determining the ratio of the height of the liquid (h) to the height of the glass (H) when the glass is half full. The volume of a cone is defined by the formula V = (1/3)πr²H. The user initially misapplied this formula, concluding that H = 2h, which is incorrect. The correct approach involves understanding the relationship between the radius and height using similar triangles to derive the correct volume ratio.

PREREQUISITES
  • Understanding of conical volume equations
  • Familiarity with the concept of similar triangles
  • Basic knowledge of mathematical modeling
  • Ability to manipulate algebraic equations
NEXT STEPS
  • Study the derivation of the volume of a cone using V = (1/3)πr²H
  • Explore the properties of similar triangles in geometric contexts
  • Learn about mathematical modeling techniques for volume calculations
  • Investigate the implications of changing dimensions on volume ratios
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Students in mathematics or engineering fields, educators teaching geometry, and anyone interested in geometric volume calculations.

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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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