Question

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

• A. True
• B. False

Answer 1

The correct option is B False

Compute the volume of the right circular cone.

Let us assume the radius and height of the right circular cone is $r$ and $h$respectively.

We know that volume of a right circular cone is given by:

$V=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

Given: Radius of the right circular cone is halved and height is doubled.

Thus, changed radius of the right circular cone $r\text{'}=\frac{r}{2}$

Changed height of the right circular cone $h\text{'}=2h$

Let, changed volume of the right circular cone is $V\text{'}$.

$\begin{array}{rcl}V\text{'}& =& \frac{1}{3}\mathrm{\pi }{\left(\mathrm{r}\text{'}\right)}^2\left(\mathrm{h}\text{'}\right)\\ & =& \frac{1}{3}\mathrm{\pi }{\left(\frac{\mathrm{r}}{2}\right)}^2\left(2\mathrm{h}\right)\\ & =& \frac{1}{6}{\mathrm{\pi r}}^2\mathrm{h}\\ & =& \frac{1}{2}V\end{array}$

We observe that the new volume is half of the original volume.

Hence the given statement is false.