Question

A solid sphere is cut into two identical hemispheres.

Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere.

Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere.

Which of the following is valid?

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true and Statement 2 is false.

• Statement 1 is false and Statement 2 is true.

Answer 1

Statement 1 is true and Statement 2 is false.

Explanation:

Statement 1: The total volume of two hemispheres is equal to the volume of the original sphere.

The volume V of a sphere with radius r is given by:

`V = 4/3πr^3`

When a sphere is cut into two hemispheres, each hemisphere will have half the volume of the original sphere.

Therefore, the volume of one hemisphere is:

`V_("hemisphere") = 1/2 xx 4/3πr^3 = 2/3πr^3`

Since there are two hemispheres, the total volume of the two hemispheres is:

This is equal to the volume of the original sphere.

Thus, Statement 1 is true.

Statement 2: The total surface area of two hemispheres together is equal to the surface area of the original sphere.

The surface area A of a sphere with radius r is given by:

A = 4πr²

When the sphere is cut into two hemispheres, each hemisphere will have:

• A curved surface area: 2πr²
• A flat circular base area: πr²

The total surface area of one hemisphere is:

`V_("hemisphere") = 2πr^2 + πr^2 = 3πr^2`

Since there are two hemispheres, the total surface area of the two hemispheres is:

2 × 3πr² = 6πr²

This is more than the surface area of the original sphere, which is 4πr².

The additional area comes from the flat circular bases of the hemispheres.

Thus, Statement 2 is false.