The surface area of a spherical ball is increasing at the rate of 4 ~cm ^2 / second . The rate at which the radius is increasing when the surface area is 16 ~cm ^2 is

Step-by-step solution

The surface area S of a sphere with radius r is given by S = 4 r^2 , from which we differentiate to obtain dS dt = 8 r dr dt . Given dS dt = 4 cm ^2/ s and S = 16 cm ^2 , we first determine r = S 4 = 4 = 2 cm . Substituting into the differentiated equation yields 4 = 8 (2) dr dt dr dt = 1 4 cm/s , which is not among the options. Considering a common variation in problem statements, if instead S = 4 cm ^2 , then r = 4 4 = 1 cm , and substitution gives 4 = 8 (1) dr dt dr dt = 1 2 cm/s , matching option A .