- #1
mburt
- 52
- 0
Hi, so this morning I made an attempt at this... With javascript (website programming language) I was able to successfully yield the ratio of the volume of a cone compared with the volume of a cylinder (1/3).
This is the source code:
And basically this is the idea. It's a summation of the area of a circle from radius 0 to 1 (adding on 0.0001 each time), where the height is the number of times the summation is iterated. So in this case it iterates 1000 times... Therefore this is the height, in units.
The volume of the cone = 10473.5 units^2
The volume of the cylinder = 31419.1 units^2
If you divide the volume of cone by cylinder you get approximately one third.
... Now I attempted represent this with calculus but I think I failed. This is as far as I got:
[tex]\sum_{0} i = i + \frac{1}{\infty}[/tex]
therefore
[tex]h = 1^\infty[/tex]
Vcone = [tex]\pi(5^\infty)^2 = 25\pi^\infty[/tex]
Vcylinder = [tex]\pi{(1)^2}1^\infty = \pi^\infty[/tex]
But clearly 25/1 doesn't equal 1/3. What am I doing wrong here? Is it possible to represent this relationship with a summation formula?
**EDIT: accidentally said "area" not volume. Whoops
This is the source code:
<script>
var pi = Math.PI,
cone_v = 0,
h = 0,
r = 1;
for (var i = 0, h = 0;i < r; i = i+0.0001, h++) {
cone_v += pi*i*i;
}
var cyl_v = pi*r*r*h;
var cone_constant = cone_v/cyl_v;
//cone_constant = 1/3
</script>
And basically this is the idea. It's a summation of the area of a circle from radius 0 to 1 (adding on 0.0001 each time), where the height is the number of times the summation is iterated. So in this case it iterates 1000 times... Therefore this is the height, in units.
The volume of the cone = 10473.5 units^2
The volume of the cylinder = 31419.1 units^2
If you divide the volume of cone by cylinder you get approximately one third.
... Now I attempted represent this with calculus but I think I failed. This is as far as I got:
[tex]\sum_{0} i = i + \frac{1}{\infty}[/tex]
therefore
[tex]h = 1^\infty[/tex]
Vcone = [tex]\pi(5^\infty)^2 = 25\pi^\infty[/tex]
Vcylinder = [tex]\pi{(1)^2}1^\infty = \pi^\infty[/tex]
But clearly 25/1 doesn't equal 1/3. What am I doing wrong here? Is it possible to represent this relationship with a summation formula?
**EDIT: accidentally said "area" not volume. Whoops
Last edited: