A spherical raindrop evaporates at a rate proportional to its surface area. The differential equation involving the rate of change of its radius r with time ' t ' is (where k is a positive constant)

Step-by-step solution

Let V be the volume and S be the surface area of the spherical raindrop of radius r . V = 4 3 r^3 and S = 4 r^2 The rate of evaporation is the rate of decrease of volume, which is given to be proportional to the surface area. dV dt -S dV dt = -k S (where k is a positive constant of proportionality) Substituting the expressions for V and S : d dt ( 4 3 r^3 ) = -k(4 r^2) 4 r^2 dr dt = -k(4 r^2) Since r 0 , dividing both sides by 4 r^2 gives: dr dt = -k dr dt + k = 0 Answer: dr dt + k = 0