Write True or False and justify your answer in each of the following: If the radius of a right circular cone is halved and height is doubled,the volume will remain unchanged.

(FALSE) Let the original radius of the cone be $r$ and the original height be $h$.
The volume of the original cone is $V_1 = \frac{1}{3} \pi r^{2} h$.
Now,the new radius $r' = \frac{r}{2}$ and the new height $h' = 2h$.
The new volume $V_2 = \frac{1}{3} \pi (r')^{2} h' = \frac{1}{3} \pi \left(\frac{r}{2}\right)^{2} (2h)$.
Simplifying this,$V_2 = \frac{1}{3} \pi \left(\frac{r^2}{4}\right) (2h) = \frac{1}{2} \left(\frac{1}{3} \pi r^{2} h\right) = \frac{1}{2} V_1$.
Since the new volume is half of the original volume,the statement is False.