+168 A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$cm, find the radius of the original bubble.
+9479 volume of sphere \(=\,\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3\)
volume of hemisphere \(=\,\frac12\,*\,\frac43\,*\,\pi\,*\, (\text{radius of hemisphere})^3\)
Plug in \(3\sqrt[3]2\) for the radius of the hemisphere.
volume of hemisphere \(=\,\frac12\,*\,\frac43\,*\,\pi \,*\,(3\sqrt[3]2)^3 \\~\\ =\,\frac23\,*\,\pi\,*\,3^3\,*\,\sqrt[3]2^3 \\~\\ =\,\frac23\,*\,\pi\,*\,27\,*\,2 \\~\\ =\,36\pi\)
The hemisphere has the same volume as the sphere, so....
volume of sphere = volume of hemisphere
\(\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3\,=\,36\pi\)
Divide both sides by pi .
\(\frac43\,*\, (\text{radius of sphere})^3\,=\,36\)
Multiply both sides by 3/4 .
\((\text{radius of sphere})^3\,=\,27\)
Take the cube root of both sides.
\(\text{radius of sphere}\,=\,3\,\text{ cm}\)
+9479 volume of sphere \(=\,\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3\)
volume of hemisphere \(=\,\frac12\,*\,\frac43\,*\,\pi\,*\, (\text{radius of hemisphere})^3\)
Plug in \(3\sqrt[3]2\) for the radius of the hemisphere.
volume of hemisphere \(=\,\frac12\,*\,\frac43\,*\,\pi \,*\,(3\sqrt[3]2)^3 \\~\\ =\,\frac23\,*\,\pi\,*\,3^3\,*\,\sqrt[3]2^3 \\~\\ =\,\frac23\,*\,\pi\,*\,27\,*\,2 \\~\\ =\,36\pi\)
The hemisphere has the same volume as the sphere, so....
volume of sphere = volume of hemisphere
\(\frac43\,*\,\pi\,*\, (\text{radius of sphere})^3\,=\,36\pi\)
Divide both sides by pi .
\(\frac43\,*\, (\text{radius of sphere})^3\,=\,36\)
Multiply both sides by 3/4 .
\((\text{radius of sphere})^3\,=\,27\)
Take the cube root of both sides.
\(\text{radius of sphere}\,=\,3\,\text{ cm}\)
