How do I prove that a cylinder and hemisphere have the same volume?
The cylinder has a radius of x and a height of h and the hemisphere has a radius of 3x/2 and x:h = 4:9
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answers:
billrussell42 say: The cylinder has a radius of x and a height of h and the hemisphere has a radius of 3x/2 and x:h = 4:9

Cylinder V = πr²h = πx²h
hemisphere V = (2/3)πr³ = (2/3)π(3x/2)³

set the two volumes equal
πx²h = (2/3)π(3x/2)³
simplify
x²h = (2/3)x³(27/8)
x²h = x³(9/4)
h = x(9/4)
h/x = 9/4
or
x/h = 4/9



Sphere V = ⁴/₃πr³
hemisphere = (2/3)πr³
Cylinder V = πr²h
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say: Volume of cylinder is V_cyl = (πr²L) = πx²h
x/h = 4/9 so h = 9x/4
V_cyl = πx²(9x/4) = (9/4)πx³

Volume of sphere = ⁴/₃πr³
Volume of sphere with r=3x/2 is ⁴/₃π(3x/2)³ = ⁴/₃π(27x³/8) = (9/2)πx³
Volume of hemisphere V_hemi = (9/4)πx³

Both volumes are (9/4)πx³ hence they are equal.
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