Question

Assertion (A): A top is of the shape of a cone mounted over a hemisphere of the same radius. If the radius of the hemisphere is 3.5 cm and the total height of the top is 15.5 cm then the total surface area of the top is $$214.5\ \mathrm{cm}^{2}$$.

Reason (R): The total surface area of the top is the sum of the curved surface area of the hemisphere and the curved surface area of the cone.

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’s height is $$15.5 - 3.5 = 12\ \mathrm{cm}$$ and its slant height is $$\sqrt{12^{2} + 3.5^{2}} = 12.5\ \mathrm{cm}$$. Thus: $$\text{Total surface area} = \pi rl + 2\pi r^{2}$$ $$= \frac{22}{7} \times 3.5 \times 12.5 + 2 \times \frac{22}{7} \times (3.5)^{2} = 214.5\ \mathrm{cm}^{2}$$

Step 2 – Reason: R is true. The joined circular faces are internal, so the exposed area is the sum of the cone’s and hemisphere’s curved surface areas.

Step 3 – Link: This is precisely the method used to calculate the total exposed surface area in A, so R correctly explains A.