Question

Assertion (A): If a conical cup has the same radius but half the height as that of a cylindrical cup then the ratio of the volume of the conical cup to that of the cylindrical cup is 2 : 3.

Reason (R): The volume of a cone is one third of the volume of a cylinder of the same base radius and same height.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Assertion (A) is false and Reason (R) is true.

Explanation:

Step 1 – Assertion: A is false. If the cylindrical cup has height $$h$$, the conical cup has height $$\frac{h}{2}$$. Their volume ratio is: $$\frac{1}{3}\pi r^{2}\left(\frac{h}{2}\right) : \pi r^{2}h = 1 : 6$$. This is not $$2 : 3$$.

Step 2 – Reason: R is true. For the same base radius and height, a cone’s volume is one third of a cylinder’s volume.

Therefore, the correct option is: Assertion (A) is false and Reason (R) is true.