The ratio of the volumes of two spheres is 1 : 8, then the ratio of their surface areas is
1. 1 : 2
2. 1 : 4
3. 1 : 8
4. None of the above
1. 1 : 2
2. 1 : 4
3. 1 : 8
4. None of the above
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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