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Height of a Cone

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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

Dec 13, 2018

#1
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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   Dec 13, 2018
edited by CPhill  Dec 13, 2018
#2
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Thanks CPhill!

CoolStuffYT  Dec 15, 2018