Answer:
Change produced in cone is greater than change produced in cylinder.
Step-by-step explanation:
Given : A cylinder and a cone start with the same radius and height.
We have to find which change produces a greater increase in volume.
We know Volume of cylinder = \(\pi r^2h\)
and Volume of cone = \(\frac{1}{3}\pi r^2h\)
Where r is radius and h is height.
Let r' and h' denotes new radius and new height
Consider Cylinder first ,
The radius of the cylinder stays the same, but the height of the cylinder is doubled.
that is r' = r and h' = 2h
Then Volume of new cylinder becomes,
Volume of cylinder = \(\pi (r')^2h'=2\pi r^2h\)
\(Change\ produced = \frac{new-original}{original}\)
that is
\(Change\ produced\ in\ cylinder = \frac{2\pi r^2h-\pi r^2h}{\pi r^2h}=1\)
Now, Consider Cone, we have,
The radius of the cone is then tripled, and the height of the cone is cut in half.
r' = 3r and h' = \(\frac{h}{2}\)
Thus, The volume of new cone becomes,
Volume of cone = \(\frac{1}{3}\pi (r')^2h'=\frac{1}{3}\pi (3r)^2\frac{h}{2}\)
\(Change\ produced = \frac{new-original}{original}\)
that is
\(Change\ produced\ in\ cone =\frac{\frac{1}{3}\pi (3r)^2\frac{h}{2}-\frac{1}{3}\pi r^2h}{\frac{1}{3}\pi r^2h} =\frac{7}{2}=3.5\)
Thus, Change produced in cone is greater than change produced in cylinder.
