Question

Assertion (A): The total surface area of a solid hemisphere of diameter $$d$$ is $$\frac{3}{4}\pi d^{2}$$.

Reason (R): If the diameter of a solid hemisphere is halved then its total surface area reduces to 25% of the original value.

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 but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Since the total surface area of a solid hemisphere is $$3\pi r^{2}$$ and $$r = \frac{d}{2}$$ : $$3\pi r^{2} = 3\pi\left(\frac{d}{2}\right)^{2} = \frac{3}{4}\pi d^{2}$$

Step 2 – Reason: R is true. Halving the diameter also halves the radius, and surface area scales with the square of the radius; hence the new area is $$\left(\frac{1}{2}\right)^{2} = \frac{1}{4}$$ of the original, or 25%.

Step 3 – Link: R describes how surface area changes when the diameter is halved; it does not establish the formula stated in A.