Question

The radius and height of a right circular cone are in the ratio 3 : 4. If its curved surface area (in cm²) is 240 π, then its volume (in cm³) is:
1. 1536 π 
2. 768 π 
3. 384 π
4. 2304 π 

Answer 1

Correct Answer - Option 2 : 768 π 

Given:

Ratio of radius and height of the cone = 3 ∶ 4

Curved surface area = 240 π cm²

Formula used:

Curved surface area of cone = π r l

Volume of cone = (1/3) π r² h

Slant height = √(h² + r²) 

Where,

r = Radius of the cone

h = Height of the cone

l = slant height

Calculations:

Let the radius and height of the cone be 3x and 4x

Slant height = √[(4x)² + (3x)²] 

⇒ √(16x² + 9x²) 

⇒ √25x2 

⇒ 5x

Curved suface area = π × 3x × 5x

⇒ 240 π = π × 15x²

⇒ x² = 16

⇒ x = 4

Radius = 12 cm

Height = 16 cm

Slant height = 20 cm

Volume of the cone = (1/3) × π × 144 × 16

⇒ 48 × 16 × π 

⇒ 768 π 

∴ The volume of the cone is 768 π cm³