Question

The surface area of a solid sphere is increased by 21% without changing its shape. Find the percentage increase in its - 1) radius , 2) volume

Answer 1

Let the original and increased surface areas be A1 and A2.

A1­/A2 = (r1/r2)2

1.21A1­/A1 = (r2/r1)2
 
1.1 = r2/r1

∴ r2 = 1.1r1
 
∴ Increase percentage = (r2 – r1) / (r1) x 100 % = (1.1r1 – r1)/ (r1) x 100 % = 10

Hence, the percentage increase in diameter is 10 %
 
Let the original and increased volume be V1 and V2.

V1/V2 = (r1­/r2)3
 
V2/V1 = (11­/10)3
 
         = 1.331
 
∴ V2 = 1.331 V1
 
Increase % = (V2 – V1) / (V1) x 100 % = (1.331 V1 – V1) / V1 x 100 %
 
                    = 33.1%
 
Hence, the percentage increase in volume is 33.1 %