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

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

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.