The rate of change of volume of spherical balloon at any instant is directly proportional to its surface area. If initially its radius is 3 cm , after 2 minutes its radius becomes 9 cm , then radius of balloon after 4 minutes is

Step-by-step solution

A spherical balloon's volume and surface area are expressed by V = 4 3 r^3 and A = 4 r^2 , respectively, for radius r . Given dV dt = kA with proportionality constant k , differentiate V : dV dt = 4 r^2 dr dt Substituting into the proportional relation: 4 r^2 dr dt = k(4 r^2) Simplify by dividing through by 4 r^2 (for r 0 ): dr dt = k This indicates a constant rate of change of the radius. Integrate with respect to t : r(t) = kt + C Given r(0) = 3 , we find C = 3 , so r(t) = kt + 3 ; with r(2) = 9 , solve 9 = 2k +