SOLUTION: A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is 12\s

Algebra ->  Surface-area -> SOLUTION: A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is 12\s      Log On


   

Question 1135970: A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is 12\sqrt{3} inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the volume, in cubic inches, of the sphere? Express your answer in terms of $\pi$.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
Since the vertex angle of the cone is 60°, the cross-section triangle is
an equilateral triangle, with all sides equal in length, and each interior
angle equaling 60°.

The volume of the sphere is V=expr%284%2F3%29%2Api%2Ar%5E3.
So we need only find the radius r and substitute.



Triangle AOB is a 30°-60°-90° right triangle, so the longer leg is √3 times
the shorter leg, so:

AB=sqrt%283%29%2AAO
12%2Fsqrt%283%29=sqrt%283%29%2Ar
Square both sides:
144%2F3=3%2Ar%5E2
48=3r%5E2
Divide both sides by 3
16=r%5E2
Take positive square root of both sides:
4=r
Substitute 4 for r in
V=expr%284%2F3%29%2Api%2Ar%5E3
V=expr%284%2F3%29%2Api%2A4%5E3
V=expr%284%2F3%29%2Api%2A64
V=expr%28256%2F3%29%2Api cubic inches  <--Volume of the sphere.

Edwin