Question

Assertion: In a cylinder, if radius is halved and height is doubled, the volume will be halved.

Reason: In a cylinder, radius is doubled and height is halved, curved surface area will be same.

Which of the following is correct?

• A. If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.
• B. If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
• C. If Assertion is correct but Reason is incorrect.
• D. If Assertion is incorrect but Reason is correct.

Answer 1

The correct option is B

If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.

Explanation for correct option

B:

Assertion:

Step 1: Given data

Radius of original cylinder$=r$

Height of original cylinder$=h$

Radius of new cylinder$=\frac{r}{2}$

Height of new cylinder$=2h$

Change in volume$=?$

Step 2: Using the formula

Volume of cylinder$={\mathrm{\pi r}}^2\mathrm{h}$

Step 3: Calculating the change in volume

Original volume of cylinder$={\mathrm{\pi r}}^2\mathrm{h}$

New volume of cylinder$=\mathrm{\pi }{\left(\frac{\mathrm{r}}{2}\right)}^2\times 2\mathrm{h}=\mathrm{\pi }\left(\frac{{\mathrm{r}}^2}{4}\right)\times 2\mathrm{h}=\frac{{\mathrm{\pi r}}^2\mathrm{h}}{2}$

Original volume of cylinder$-$New volume of cylinder$={\mathrm{\pi r}}^2\mathrm{h}-\frac{{\mathrm{\pi r}}^2\mathrm{h}}{2}={\mathrm{\pi r}}^2\mathrm{h}\left(1-\frac{1}{2}\right)=\frac{1}{2}{\mathrm{\pi r}}^2\mathrm{h}$

Thus, in a cylinder, if radius is halved and height is doubled, the volume will be halved.

Assertion is correct.

Reason:

Step 1: Given data

Radius of original cylinder$=r$

Height of original cylinder$=h$

Radius of new cylinder$=2r$

Height of new cylinder$=\frac{h}{2}$

Change in curved surface area of cylinder$=?$

Step 2: Using the formula

Curved surface area of cylinder$=2\mathrm{\pi rh}$

Step 3: Calculating the change in curved surface area

Original CSA of cylinder$=2\mathrm{\pi rh}$

New CSA of cylinder$=2\mathrm{\pi }\times 2\mathrm{r}\left(\frac{h}{2}\right)=2\mathrm{\pi rh}$

Original CSA of cylinder$=$New CSA of cylinder

Thus, in a cylinder, radius is doubled and height is halved, curved surface area will be same.

Reason is correct.

However, Reason has no relation to the Assertion. Thus, Reason is not the correct explanation of Assertion.

Hence, option B is correct.