Question

If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved. Write True or False and justify your answer in the following.

• A. True
• B. False

Answer 1

The correct option is A True

Step 1: State the given data and formula used

Let, the original radius of the cylinder be $rcm$ and the original height be $hcm$.

The formula for the curved surface area of the cylinder $=2\pi rh$

For the new cylinder,

New Radius, $r_1=2rcm$

New height be, $h_{1}cm$

Area of new cylinder $=2{\mathrm{\pi r}}_1{\mathrm{h}}_1$

Step 2: Compare the curved surface area:

Now, according to the question, the curved surface area is not changed. So,

$$\begin{gathered} 2\pi rh=2{\mathrm{\pi r}}_1{\mathrm{h}}_1 \\ \Rightarrow rh=r_1{\mathrm{h}}_1 \\ \Rightarrow rh=2r{h}_1\mathbf{[}{\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{2}\mathit{r}\mathbf{]} \\ \Rightarrow h=2h_1 \\ \Rightarrow h_1=\frac{h}{2} \end{gathered}$$

Hence, it is proved that height must be halved to have the same curved surface area when the radius is doubled.

Therefore the given statement is True.