A spherical rain drop evaporates at a rate proportional to its surface area. If initially its radius is 3 mm and after 1 second it is reduced to 2 mm , then at any time t its radius is (where 0 t 3 )

Step-by-step solution

aligned & dv dt - s & dv dt =- ks , ...(i) & where k 0 & v = 4 3 r ^3 and s =4 r ^2 & dv dt =4 r ^2 dr dt aligned Equation (i) becomes aligned & 4 r ^2 dr dt =- k (4 r ^2 ) & dr dt =- k aligned Integrating on both sides, we get array ll & r=-k t+c...(ii) & When t=0, r=3 & 3=-k(0)+c c=3 & r=-k t+3 ...[From(ii)] & When t=1, r=2 & 2=-k(1)+3 k=1 & r=-t+3 & r=3-t array