A spherical balloon is being inflated at a constant rate of 17 cubic inches per second How fast is th radius of the balloon changing at the instant the balloon s diameter is 8 inches Is the radius changing more rapidly when d 8 or when d 12 Why equation V 4 Recall that the volume of a sphere of radius r is V Tr Note well that in the setting of this 3 problem both V and r are changing as time t changes and thus both V and r may be viewed as dV dr Differentiate both sides of the implicit functions of t with respective derivatives and dt dt 4 3 Tr with respect to t using the chain rule on the right to find a formula for dr dt dV dt Time 1 that depends on both r and dV dt Time 2 dr dt At this point in the problem by differentiating we have related the rates of change of V and r Recall that we are given in the problem that the balloon is being inflated at a constant rate of 17 cubic inches per second Is this rate the value of dr dv dt dt Time 3 or dV dt