SOLUTION: A sector of 120 degrees has a radius of 9 cm. The sector is then folded to form a cone. Find the exact value for the height "h" of the cone. Hence, find the volume.

Algebra ->  Volume -> SOLUTION: A sector of 120 degrees has a radius of 9 cm. The sector is then folded to form a cone. Find the exact value for the height "h" of the cone. Hence, find the volume.       Log On


   

Question 1109048: A sector of 120 degrees has a radius of 9 cm. The sector is then folded to form a cone. Find the exact value for the height "h" of the cone. Hence, find the volume.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
120%5Eo%2F360%5Eo=1%2F3
The sector is 1%2F3 of the circle,
so its arc is 1%2F3 of the circumference of radius 9cm .
When the sector is folded, that arc will be
the circumference of the base of the cone,
so that circumference will be 1%2F3 of the circumference of radius 9cm ,
and its radius will be 1%2F3 of 9cm ,
or r%5Bcone%5D=%281%2F3%29%289cm%29=3cm .
The slant height of the cone formed is the 9cm radius of the sector.
A cross section of the cone passing through apex and center of the base will look like this
As per Pythagorean theorem, the exact value of the height is
h=sqrt%28%289cm%29%5E2-%283cm%29%5E2%29%22=%22sqrt%2872%29cm%22=%226sqrt%282%29cm .
An approximate value is 8.485cm
Then, the volume is
V%5Bcone%5D%22=%22%28pi%2F3%29h%2Ar%5E2%22=%22%28pi%2F3%296sqrt%282%293cm%5E3%22=%2218sqrt%282%29%2Apicm%5E3 .
That is the exact expression for the volume.
79.97cm%5E3 is an approximate value.