How Do You Derive the Volume Formula for a Cone with Variable Radius and Height?

[chris777]

I can't figure out how to derive this. This is the formula for the volume of a cone. R is a constant and the side of the cone. Can be any real number.

If anyone could take a crack at deriving this id be very greatful!

$$V = \pi/3 * (R(1 - x/2\pi))^2 * \sqrt{(R^2 - (R(1 - x/2\pi))^2}$$

 

[Jameson]

Is that your formula or the one you need to derive? And what exactly are you tring to derive? The formula for the volume of a cone?

 

[dextercioby]

$$V=\frac{\pi r^{2}h}{3}$$

.If the (semi)cone is rectangular (the axis joining the top and the center of the base (assumed a circle)),then u can use Pythagora's theorem

$$R^{2}=r^{2}+h^{2}$$ and then can express the volume in terms of "R" and either the height "h",or the radius of the circle (the base) "r".

Daniel.

 

[chris777]

r = R(1-x/2pi)
h = sqrt( R^2 - ( R (1-x/2pi)))

these are put into the formula for the volume of the cone. Now I need to derive that equation to know which x will give the max volume of the cone.