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A cylinder is inscribed in a sphere of radius 1. Express the volume V of the cylinder as a function of the base radius r.
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Volume of a cylinder is
V = , (1)
where r is the radius of the cylinder and h is the height of the cylinder.
When cylinder of the radius r is inscribed in a sphere of radius r,
h = (2)
(make a sketch and apply the Pythagorean theorem).
Substitute (2) into (1), and you will get
V = .
This is the required formula.
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Comment from student: Thanks. Please show me how you derive the height algebraically.
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My responce
It is not algebraically.
It is geometrically.
Make a sketch of the section of the sphere with the cylinder inscribed.
Then apply the Pythagorean Theorem.
It is so obvious.