Question

The largest possible right circular cone is carved out of a solid hemisphere of radius ‘r’ as shown in the figure below. The slant height of the cone is:

पर्याय

• r

• 2r

• `sqrt(2)r`

• `sqrt(3)r`

Answer 1

`bb(sqrt(2)r)`

Explanation:

1. Identify the cone’s dimensions

For a right circular cone to have the largest possible volume when carved from a hemisphere, its base must lie on the flat circular base of the hemisphere.

The radius of the cone’s base (R) is equal to the radius of the hemisphere, so R = r.

The height of the cone (h) is equal to the radius of the hemisphere because the vertex touches the top of the hemisphere, so h = r.

2. Calculate the slant height

The slant height (l) of a right circular cone is the hypotenuse of the right-angled triangle formed by its radius and height. 

Using the Pythagorean Theorem:

l² = R² + h²

Substitute R = r and h = r:

l² = r² + r²

l² = 2r²

`l = sqrt(2r^2)`

`l = sqrt(2)r`