SOLUTION: A sector is folded to form a cone. If the radius of the cone is 3cm, find a) the exact height of the cone if the total surface area is 72π cm squared. b) radius of the sect

Algebra ->  Surface-area -> SOLUTION: A sector is folded to form a cone. If the radius of the cone is 3cm, find a) the exact height of the cone if the total surface area is 72π cm squared. b) radius of the sect      Log On


   

Question 1121711: A sector is folded to form a cone. If the radius of the cone is 3cm, find
a) the exact height of the cone if the total surface area is 72π cm squared.
b) radius of the sector
c) angle of the sector folded
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The total surface area is the area of the base plus the lateral surface area:

%28pi%29r%5E2%2B%28pi%29%28r%29%28l%29

where l is the slant height.

The radius of the cone is 3; and the total surface area is 72pi. We can plug those numbers into the formula for the total surface area to determine the slant height; then the Pythagorean Theorem will give us the height of the cone.

9%28pi%29%2B%28pi%29%283%29%28l%29+=+72%28pi%29
3%28pi%29l+=+63%28pi%29
l+=+21

3%5E2%2Bh%5E2+=+21%5E2
h%5E2+=+441-9+=+432
h+=+12%2Asqrt%283%29

(a) The height of the cone is 12*sqrt(3)cm.

(b) The radius of the sector is the slant height of the cone: 21 cm.

The circumference of the base of the cone is the arc length of the sector. The fraction that arc length is of the entire circle is the ratio of the circumference of the cone to the circumference of the circle:

6%28pi%29%2F42%28pi%29+=+1%2F7

(c) The angle of the sector is 1/7 of the full circle: 2pi/7.