Prove that cone is 1/3 of the cylinder
Answers
Answer: mathematically and experimentally
Step-by-step explanation:
MATHEMATICALLY
To get the answer of above question lets calculate the volume and cylinder and volume of cone separately and deduce a relationship between them. I would be using Calculus for the same.
Lets Assume for both Cylinder and Cone:
Radius of Cylinder/Cone : RR
Height of Cylinder/Cone : HH
1. Volume of Cylinder
Basically, a cylinder is composed of several discs, each of same radius (RR) placed one over the other.
Lets assume that, each disc is of height “dh”“dh”
So, to get its volume we need to integrate the area of discs over its height (HH).
Area of Single disc:
A=πR2A=πR2
Volume of Cylinder:
V=∫H0πR2dhV=∫0HπR2dh
=πR2∫H0dh=πR2∫0Hdh
V=πR2HV=πR2H
Volume of Cylinder=πR2H=πR2H
1. Volume of Cone
We can assume the cone to be constructed of disks of infinitesimal thickness stacked one on top of the other, with the largest disk having radius RR at height h=Hh=H and the smallest having radius00at height h=0h=0.
Radius of largest Disk :R(:R(at height h=H)h=H)
Radius of Smallest Disk:0(:0(at height h=0)h=0)
The radius of any disk in between, “rr” can be written as
r=R(h/H)r=R(h/H)
The volume of a single disk, dVdV is
dV=πR2dh=πR2.(H/R)drdV=πR2dh=πR2.(H/R)dr
and the volume of the cone is then obtained by integrating(summing) the volume of each of the disks
V=∫dVV=∫dV
V=∫πR2(H/R),drV=∫πR2(H/R),dr
V=πH/R∫R0πR2drV=πH/R∫0RπR2dr
V=1/3πR2HV=1/3πR2H
Volume of Cone=1/3πR2H=1/3πR2H
So from the above two derivations, we can say that
Volume of Cone = 1/3 of Volume of Cylinder
EXPERIMENTALLY
Materials Required
One cone and one cylinder having the same height and base radius, sand.
Procedure
Fill the cone with sand.
Pour the sand from the cone to the cylinder.
Fill the cone with sand again and pour to the cylinder.
Repeat the same process until the cylinder fills completely with sand.
cbse-class-10-maths-lab-manual-volume-of-a-cone-1
Observation and Result
Students will observe that the cylinder gets filled after pouring the sand three times from cone.