The ratio of the volumes of two spheres is 1 : 8, then the ratio of th

1.	The ratio of the volumes of two spheres is 1 : 8, then the ratio of their surface areas is 1. 1 : 22. 1 : 43. 1 : 84. None of the above
Answer» Correct Answer - Option 2 : 1 : 4 Given: The ratio of volume of two spheres = 1 ∶ 8 Formula used: The volume of sphere = (4/3) × π r³ The surface area of sphere = 4π r² Calculations: Let us take the radius of 1st sphere to be r1 Let us take the radius of the 2nd sphere to be r2 The volume of 1st sphere to be v₁ = (4/3) × π r₁3 The volume of 1st sphere to be v1 = (4/3) × π r₂3 The ratio of volume the two spheres = \(\)[ (4/3) × π r₁³] ÷ [(4/3) × π r23] ⇒ [ (4/3) × π r13] ÷ [(4/3) × π r23] = 1 ∶ 8 ⇒ (r₁÷ r₂₎³ = 1/8 ⇒ (r1 ÷ r2)3 = (1/2)³ ⇒ r1 ÷ r2 = 1/2 The surface area of 1st sphere = 4π r₁2 The surface area of 2nd sphere = 4π r₂2 The ratio of surface area of two spheres = (4π r12) ÷ (4π r22) ⇒ (r₁÷ r₂)² = (1/2)² ⇒ 1/4 ∴ The ratio of surface area of two spheres = 1 ∶ 4

The ratio of the volumes of two spheres is 1 : 8, then the ratio of their surface areas is 1. 1 : 22. 1 : 43. 1 : 84. None of the above

Answer» Correct Answer - Option 2 : 1 : 4

Given:

The ratio of volume of two spheres = 1 ∶ 8

Formula used:

The volume of sphere = (4/3) × π r³

The surface area of sphere = 4π r²

Calculations:

Let us take the radius of 1st sphere to be r1

Let us take the radius of the 2nd sphere to be r2

The volume of 1st sphere to be v₁ = (4/3) × π r₁3

The volume of 1st sphere to be v1 = (4/3) × π r₂3

The ratio of volume the two spheres = \(\)[ (4/3) × π r₁³] ÷ [(4/3) × π r23]

⇒ [ (4/3) × π r13] ÷ [(4/3) × π r23] = 1 ∶ 8

⇒ (r₁÷ r₂₎³ = 1/8

⇒ (r1 ÷ r2)3 = (1/2)³

⇒ r1 ÷ r2 = 1/2

The surface area of 1st sphere = 4π r₁2

The surface area of 2nd sphere = 4π r₂2

The ratio of surface area of two spheres = (4π r12) ÷ (4π r22)

⇒ (r₁÷ r₂)² = (1/2)²

⇒ 1/4

∴ The ratio of surface area of two spheres = 1 ∶ 4