volume of a cone is equal to one third

VOLUME OF A CONE IS EQUAL TO ONE-THIRD OF THE VOLUME OF A CYLINDER

OF THE SAME BASE AND HEIGHT.MATHEMATICS HOLIDAY HOMEWORK

2014-2015

QUESTION TO BE ANSWERED

Verify that volume of a cone is equal to one-third of the

volume of a cylinder of the same base and

height.

RELATION OF A CONE AND A CYLINDER

ANSWER

Using Calculus: 

Firstly, the cylinder is easy, just have lots of little discs, each disc has a radius of "r" and an area of πr², and we need to integrate over height: 

V = ∫πr² dh = πr²h 

That was fairly easy! 

Now for the cone, the little disc's radius get smaller as you get higher. The rate they get smaller is a constant, the slope of the sides, which we can call s. 

For simplicity I will turn the cone upside-down, so the disc's radius get bigger with height. Each disc will have a radius of "sh" and an area of π(sh)², and we need to integrate over height: 

Which is one third of the cylinder's volume! 

Now, I recall seeing proofs without calculus that did something similar, they turn the cone into "N" small discs, and add them up. 

Disc number "i" will have a height of h/N, and a radius of r(i/N), with a volume of πr²(i/N)²(h/N) = πr²i²h/N³ 

Summing over "i" : V = ∑πr²i²h/N³ = πr²h/N³ ∑i² 

The only thing in the way now is "∑i²", which I know can be simplified to a few terms, but am not sure how. If that could be done, hopefully we would see the formula come out to be πr² (1/3)h 

So, I haven't totally solved this for you, but I hope I have helped.

V = ∫π(sh)² dh = ∫πs²h² dh = πs² (1/3)h³ 

Now we know that the slope s = r/h, so: V = π(r/h)² (1/3)h³ = πr² (1/3)h 

This or turn to the next slide for more

Thank You

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