+1253 A cone has a volume of 12288pi cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone?
I've got that you can split the cone into 30-60-90 triangles, but I don't what to do after that.
Thank you very much!
:P
+128399 Your approach is the correct one....!!!!
Let's call the radius of the cone, R
If we bisect the vertex angle, we get a 30-60-90 right triangle
The side across from the 30° angle = R
The side across from the 60° angle = R*sqrt (3) = cone height
So
Vcone = (1/3)pi R^2 * height
Vcone = (1/3)pi* R^2 * R*sqrt (3) ......so we have
12288pi = (1/3) pi* R^3 * sqrt(3)
12288 = [sqrt(3)/ 3] R^3 multiply both sides by 3/sqrt(3
[ 12288* 3 / sqrt(3) ] = R^3
[ 12288 sqrt(3) *sqrt(3) / sqrt(3) ] = R^3
[ 12288 sqrt (3) ] = R^3 Note : { 12288 = 2^12 * 3 }
[ 2^12 * 3 sqrt(3) ] = R^3
[ 2^12 * 3^(3/2) ] = R^3 take the cube root of both sides
2^4 * 3^1/2 = R
16sqrt(3) = R
So....the height is R*sqrt(3) = 16sqrt(3) * sqrt (3) = 16 * 3 = 48in
