Question

Assertion (A): A solid is in the form of a cone of radius $$r$$ surmounted on a hemisphere of the same radius. If the height of the cone is the same as the diameter of its base then the volume of the solid is $$\frac{4}{3}\pi r^{3}$$.

Reason (R): Total volume of the solid $$=\text{volume of the cone}+\text{volume of the hemisphere}$$

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

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

Explanation:

Step 1 – Assertion: A is true. The cone height is $$2r$$, so: $$V = \frac{1}{3}\pi r^{2}(2r) + \frac{2}{3}\pi r^{3}$$ $$V = \frac{4}{3}\pi r^{3}$$

Step 2 – Reason: R is true. The total volume of the composite solid is the sum of the volumes of the cone and hemisphere.

Step 3 – Link: Adding those two component volumes, using the stated cone height, gives the volume in A; therefore, R correctly explains A.