The diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease?
Let the diameter of the sphere be “d”.
Radius of sphere, r1 = \(\frac{d}{2}\)
New radius of sphere, say r2 = \(\frac{d}{2}\) x \(\frac{1 - 25}{100}\)
= \(\frac{3d}{8}\)
Curved surface area of sphere, (CSA)1
= 4πr12 = 4π × (d/2)2 = πd2 …(1)
Curved surface area of sphere when radius is decreased (CSA)2 = 4πr22
= 4π×(3d/8)2 = (9/16)πd2 …(2)
From equation (1) and (2), we have
Decrease in surface area of sphere = (CSA)1 – (CSA)2
= πd2 – (9/16)πd2
= (7/16)πd2
Percentage decrease in surface area of sphere
= \(\frac{(CSA)1 – (CSA)2}{(CSA)1}\) x 100
= (7d2/16d2) × 100 = \(\frac{700}{16}\) = 43.75%
Therefore, the percentage decrease in the surface area of the sphere is 43.75%.