Question 1151079
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Vs = Volume of sphere
Vs = (4/3)*pi*r^3


A hemisphere is half a sphere.
Vh = Volume of hemisphere
Vh = (1/2)*Vs
Vh = (1/2)*(4/3)*pi*r^3
Vh = (4/6)*pi*r^3
Vh = (2/3)*pi*r^3


We're told that both the hemisphere and cone have the same height. 
The height of the hemisphere is the radius r, so for the cone, h = r.
Vc = Volume of cone
Vc = (1/3)*pi*r^2*h
Vc = (1/3)*pi*r^2*r ... plug in h = r
Vc = (1/3)*pi*r^3


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Divide the hemisphere volume over the cone volume.


{{{Vh/Vc = ((2/3)*pi*r^3)/((1/3)*pi*r^3)}}}


{{{Vh/Vc = ((2/3)*cross(pi)*r^3)/((1/3)*cross(pi)*r^3)}}} The pi terms cancel.


{{{Vh/Vc = ((2/3)*r^3)/((1/3)*r^3)}}}


{{{Vh/Vc = ((2/3)*cross(r^3))/((1/3)*cross(r^3))}}} The r^3 terms cancel.


{{{Vh/Vc = (2/3)/(1/3)}}}


{{{Vh/Vc = (2/3)*(3/1)}}} Flip the second fraction and multiply.


{{{Vh/Vc = (2*3)/(3*1)}}}


{{{Vh/Vc = 6/3}}}


{{{Vh/Vc = 2}}}


<font color=red size=4>The ratio of the hemisphere volume to the cone volume is 2:1. </font>


This means the hemisphere has twice the volume of the cone.
Put another way,
Vh = 2*Vc
which can be rearranged to
Vc = (1/2)*Vh


This only works if the cone and hemisphere share the same circular base, and also have the same height (h = r).


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side note:


If we start with Vh = 2*Vc and plug in Vc = (1/3)*pi*r^3, then we get,
Vh = 2*Vc
Vh = 2*(1/3)*pi*r^3
Vh = (2/3)*pi*r^3
Or we could start with Vc = (1/2)*Vh and plug in Vh = (2/3)*pi*r^3 to get,
Vc = (1/2)*Vh
Vc = (1/2)*(2/3)*pi*r^3
Vc = (1/3)*pi*r^3
This helps confirm our answer.
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