you have to prove that vol of a cylind is thrice the vol of a cone
2006-12-30 15:58:31 · 7 answers · asked by madhu g 1 in Science & Mathematics ➔ Mathematics
Are you allowed to use the well-known formulas for the volumes?
Cylinder:
V = pi * r^2 * h
Cone:
V = (1/3) * pi * r^2 * h
2006-12-30 16:01:17 · answer #1 · answered by ? 6 · 0⤊ 1⤋
You can use calculus to prove it directly.
Let a cone have a radius r and height h.
Put the cone's vertex at the origin such that it opens up.
We can write an equation of the side of the cone:
y = (h/r)x
The volume of the cone then can be calculated,
V
= ∫pi x^2 dy [y: 0...h]
= ∫pi (r/h)^2 y^2 dy [y: 0...h]
=(1/3) pi r^2 h
=(1/3) Volume of a cylinder with the same radius and height.
2006-12-30 16:14:40 · answer #2 · answered by sahsjing 7 · 0⤊ 0⤋
The formula for the volume of a cylinder is:
V = pi(r)squared x h
the formula for the volume of a cone is:
V = 1/3 pi(r) squared x h
therefore it can be seen that the volume of a cylinder is 3 times that of a cone
Example: a cylinder of diameter 10 and height of 20
a cone with diameter 10 and height of 20
Cylinder volume = 3.1416 x 10 x 10 x 20 = 628.32
Cone volume = 1/3 (3.1416 x 10 x 10 x20) = 209.44
209.44 x 3 = 628.32 (three times the volume
2006-12-31 18:31:45 · answer #3 · answered by David C 2 · 0⤊ 0⤋
The area of a thin slice is pi r*2. The volume of a slice of thickness dy is then pi r^2 dy. Let the cone be standing on its tip, such that the radius r at any height y is m y. Then we take the definite integral from 0 to the height h of the cone of: pi m^2 y^2 dy, which is pi m^2 h^3/3. It can also be done geometrically, but the calculus proof is idiot simple.
2006-12-30 16:17:46 · answer #4 · answered by Anonymous · 0⤊ 0⤋
for the same radius and height volume of the cylinder=pir^2h and volume of the cone=1/3 pir^2h
so the volume of the cylinder is thrice that of the cone
2006-12-30 16:02:37 · answer #5 · answered by raj 7 · 0⤊ 0⤋
I am going to assume that you mean of the same height and radius.
The formulas are:
Cylinder: pi r² * H =Vc
Cone 1/3 pi r² * H = Vcn
So if r and h are equal, then you can factor then out and you are left with
Vc=1/3Vcn
2006-12-30 16:21:55 · answer #6 · answered by Walking Man 6 · 0⤊ 0⤋
you may simply refer to the stated formulas,...
I've looked for a proof for you... here is a website which will probably help if you really want it,...
but the proof is a bit calculus based,... so I hope that is what you are really looking for.
2006-12-30 16:05:21 · answer #7 · answered by beanie_boy_007 3 · 0⤊ 0⤋