SOLUTION: A cone is circumscribed by a hemisphere radius of which is equal to the height of the cone. Calculate the ratio of the volume of the cone to the volume of the hemisphere.

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Question 1135910: A cone is circumscribed by a hemisphere radius of which is equal to the height of the cone. Calculate the ratio of the volume of the cone to the volume of the hemisphere.
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

This is what the side view looks like

The radius of the cone is r, which is also the same radius of the hemisphere. A hemisphere is half a sphere.
The height of the cone is the radius as well, so h = r

Volume of Cone = (1/3)*pi*r^2*h
Volume of Cone = (1/3)*pi*r^2*r ........ replace h with r
Volume of Cone = (1/3)*pi*r^3
Let's denote this to be C, so
C = (1/3)*pi*r^3

Volume of Sphere = (4/3)*pi*r^3
Volume of hemisphere = (1/2)*(volume of sphere)
Volume of hemisphere = (1/2)*((4/3)*pi*r^3)
Volume of hemisphere = (2/3)*pi*r^3
Let's call this D, so,
D = (2/3)*pi*r^3

Divide the expressions for C and D
Ratio = C/D
Ratio = ( C )/( D )
Ratio = ( (1/3)*pi*r^3 ) / ( (2/3)*pi*r^3 )
Ratio = (1/3) / (2/3) ......... note the "pi*r^3" terms cancel
Ratio = (1/3) * (3/2)
Ratio = (1*3)/(3*2)
Ratio = 3/6
Ratio = 1/2

The ratio of the volume of the cone to the volume of the hemisphere is 1/2, meaning the cone has half the volume of the hemisphere

You can write the ratio as 1:2 simply indicating the cone's volume is 1 part and the hemisphere's volume is 2 parts (twice as large as the "1 part")
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
The hemisphere of the radius "r" has the volume


     = .


The cone of the base radius "r" and the height "r" has the volume


     =  = .


The ratio of the two volumes is    =  =  = .       ANSWER

Solved.


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