reasing when the dameter is 40 cm ?SOUTHON We start by identilying two things:the grves information:The rate of increase of the volume of air is 150cm3s.and the unknown:The rate of increase of the radius when the diameter is 40 cm .In order to express these quantities mathematically, we introduce some suggestive notation:Let V be the woldine of the balloon and let r be its radius.The key thing to remember is that rates of change are deriyatives. In this problem, the volume and the rafius are bots functions of the time. The rate chinciecer the volume with respect to time is the derivative dV/at, and the rate of increase of the radius is dy/d.. We can therefore restate the glven and the untinen an follows:Given: ,Ndt=150cm3sUnknown: ddt ehen r=bm.then an mast newIn order to connect olydt and dryte, weinstrestel fand f by the formula for the valume of a sphere:V=43r3.In order to use the given information, we differentiate each side of this equation with respect to r . To differentite the right sile, we need to was the churn file:dvdt-dvdrdrdt=,drdtNow we solve for the unknown quantity:drdt=dVdtIf we put r=20 and ofyet =150 in this equation, we obtaindrdt=1150=The radius of the bulloon is increasing at the rate of * arr/s (rewnsed to foor decimal priacel).