Question

Assertion (A): The radius of a hemisphere increases from r cm to 2r cm. The ratio between the surface areas of the original hemisphere and the resulting hemisphere is 1 : 4.

Reason (R): Surface area in the first case = πr² + 2πr² and surface area in the second case = π(2r)² + 2π(2r)².

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

The total surface area of a hemisphere is given by πr² + 2πr² = 3πr² . For radius r, the area is 3πr², and for radius 2r, the area is 3π(2r)² = 12πr². The ratio of the original area to the resulting area is \[\frac{3\pi r^2}{12\pi r^2} = \frac{1}{4}\] (or 1 : 4). Both Assertion A and Reason R are true, with R correctly outlining the surface area formulas.