How do I calculate the volume of water in a partially filled cone?

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SUMMARY

The discussion revolves around calculating the volume of water in a partially filled cone, specifically in a scenario where a pipe is sloped and filled with water. The user initially visualizes the problem as a cone but is challenged by the mathematics of a partial cone. Another participant suggests that the problem may be more accurately represented as a cylinder intersected by a plane, indicating that the volume calculation should focus on the space between the intersection points within the cylinder. This clarification shifts the perspective from a conical to a cylindrical volume calculation.

PREREQUISITES
  • Understanding of geometric shapes, specifically cones and cylinders.
  • Familiarity with volume calculation formulas for cones and cylinders.
  • Basic knowledge of calculus, particularly regarding volume of revolution and cross-sections.
  • Ability to visualize geometric intersections and their implications on volume.
NEXT STEPS
  • Study the formula for the volume of a cone and how to calculate partial volumes.
  • Learn about the volume of a cylinder and how to determine the volume of a segment cut by a plane.
  • Explore calculus concepts related to volume of revolution and integration techniques.
  • Investigate practical applications of these volume calculations in fluid dynamics and engineering.
USEFUL FOR

Mathematicians, engineers, and students studying fluid dynamics or geometry who need to calculate volumes of irregular shapes, particularly in practical applications involving partially filled containers.

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Imagine that I have a pipe that is on a constant slope at any percent. On the low end of the pipe, there is a container filled with water. The water has naturally found a leveling point up into the pipe. Provided the level of water in the container is above the top of the pipe, a portion of the pipe will be completely filled with water. Out certain distance, the pipe will go from being completely filled to partially filled (where the top of the pipe is higher then the level of water) How can i find the volume of water that is contained in the part of the pipe that is not completely filled. The best way that I can picture this is a cone that has been laid on its side so that the center of the base, and the point of the cone are level, and partially filled with water. The cone volume is simple, but the partial cone problem is beyond my mathmatical knowledge. Any help would be beneficial to me.
 
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From your description, I don't see how a cone picture applies.

Isn't it more like a cylinder with a plane cutting through it? You would be interested in the volume contained between the top and bottom points of intersection within the cylinder.
 

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