Volume and plane intersecting sphere

AI Thread Summary
A plane tangent to a sphere is tilted at an angle theta, causing it to intersect the sphere and divide its volume into two regions. The volume of the hemisphere created by this intersection can be expressed using a specific equation involving theta. To prove this volume calculation, integration of the sphere's sections is suggested as a method. However, there is a discussion about the need for a geometric proof, particularly for those without calculus knowledge. Understanding the relationship between the plane's angle and the sphere's volume is crucial for solving the problem.
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Homework Statement



A plane is tangent to the surface of a sphere. You then tilt the plane at an angle theta along one axis, causing it to begin passing through the sphere and splitting the sphere's volume into two regions. I claim the volume of the hemisphere the plane has just passed through is found from the equation below. How do I prove this using geometry?

Homework Equations



<br /> V = \frac{\pi R^{3}}{3}( cos^{3}(\theta) - 3 cos(\theta) + 2 )<br />

The Attempt at a Solution

 
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Attach a picture please, to show what theta is.

ehild
 
You prove it with integration of the sphere sections.
I'm not sure what d'you mean "geometrically".
 
Quinzio said:
You prove it with integration of the sphere sections.
I'm not sure what d'you mean "geometrically".
I think he meant geometrically because he has not yet a working knowledge of calculus since this is in the precalculus section.
 

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