SOLUTION: From a solid wooden cone of height 20cm and slant height 25cm, largest possible hemisphere is carved. a) Find the radius of the hemisphere b) Find the volume of the remaining po

Algebra ->  Volume -> SOLUTION: From a solid wooden cone of height 20cm and slant height 25cm, largest possible hemisphere is carved. a) Find the radius of the hemisphere b) Find the volume of the remaining po      Log On


   

Question 1100756: From a solid wooden cone of height 20cm and slant height 25cm, largest possible hemisphere is carved.
a) Find the radius of the hemisphere
b) Find the volume of the remaining portion
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
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So from the diagram,
alpha%2Bbeta%2B90=180
alpha%2Bbeta=90
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cos%28alpha%29=r%2FCX=r%2F%2820-r%29
and
sin%28alpha%29=20%2F25=4%2F5
You also know that,
sin%5E2%28alpha%29%2Bcos%5E2%28alpha%29=1
16%2F25%2Br%5E2%2F%2820-r%29%5E2=1
r%5E2%2F%2820-r%29%5E2=9%2F25
r%2F%2820-r%29=3%2F5
5r=3%2820-r%29
5r=60-3r
8r=60
Solve for r.
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The volume of the cone is,
V%5Bc%5D=pi%2AR%5E2%2Ah
Calculate R using the Pythagroean theorem,
20%5E2%2BR%5E2=25%5E2
R=sqrt%28625-400%29
and the volume of the sphere is,
V%5Bs%5D=%284%2F3%29pi%2Ar%5E3 with r calculated above.
So the left over volume would be,
V=V%5Bc%5D-V%5Bs%5D