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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.

AdminMod2  Sep 24, 2017

Best Answer 

 #1
avatar+7067 
+1

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.

\(\text{radius of sphere}\,=\,3\,\text{ cm}\)

hectictar  Sep 24, 2017
 #1
avatar+7067 
+1
Best Answer

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.

\(\text{radius of sphere}\,=\,3\,\text{ cm}\)

hectictar  Sep 24, 2017
 #2
avatar+86859 
+1

 

Nice, hectictar.....!!!!

 

cool cool cool

CPhill  Sep 24, 2017

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