Shortcut Trick
Surface Area Ratio (S1:S2) = (Radius Ratio)2 = 4 : 25
Radius Ratio (r1:r2) = √4 : √25 = 2 : 5
Volume Ratio (V1:V2) = (Radius Ratio)3 = 23 : 53
Required Ratio = 8 : 125
∴ The correct option is C.
Alternate Method
Given:
Ratio of surface areas of two spheres = 4 : 25
Formula Used:
Surface area of a sphere = 4 × π × r2
Volume of a sphere = (4/3) × π × r3
r1
Area = 4k
r2
Area = 25k
Calculations:
⇒ Let the radii of the two spheres be r1 and r2.
⇒ (4 × π × r12) ÷ (4 × π × r22) = 4 ÷ 25
⇒ r12 ÷ r22 = 4 ÷ 25
⇒ r1 ÷ r2 = √(4 ÷ 25) = 2 ÷ 5
⇒ Volume Ratio = [(4/3) × π × r13] ÷ [(4/3) × π × r23]
⇒ Volume Ratio = (r1 ÷ r2)3
⇒ Volume Ratio = (2 ÷ 5)3 = 8 ÷ 125
∴ The correct option is C.
Additional Information
Properties of Sphere and Hemisphere
For a sphere of radius r, Surface Area = 4πr2 and Volume = (4/3)πr3. For a hemisphere, Curved Surface Area = 2πr2 and Total Surface Area = 3πr2.
Scaling Rules for 3D Shapes
If the linear dimensions (like radius) are in ratio a:b, then areas are in ratio a2:b2 and volumes are in ratio a3:b3. This applies to all similar solids like cubes, spheres, and cylinders.
Direct Relation between Area and Volume
The ratio of volumes of two spheres can be calculated from the ratio of their surface areas (S1/S2) using the formula: V1/V2 = (S1/S2)3/2.