A spherical balloon is being inflated at a constant rate of 20 cubic inches per second. How fast is the radius of the balloon changing at the instant the balloon's diameter is 12 inches? Is the radius changing more rapidly when d = 12 or when d = 16? Why? Draw several spheres with different radii, and observe that as volume changes, the radius, diameter, and surface area of the balloon also change. Recall that the volume of a sphere of radius r is V = ar³. Note that in the setting of this problem, both V and r are changing as time t changes, and thus both V and r may be viewed as implicit functions of t, with respective derivatives dV and Differentiate both sides of the equation V = ar with respect to t (using the chain rule on the right) to find a formula for depends on both r and dr dr that dV 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 20 cubic inches per second. To which derivative does this rate correspond? AP А. dr AP dt dr С. dt OD. None of these From the above discussion, we know the value of dV at every value of t. Next, observe that when the diameter of the balloon is 12, we dt AP 4xr2 dr dt , substitute these values for the relevant quantities and solve for the dt know the value of the radius. In the equation remaining unknown quantity, which is d. How fast is the radius changing at the instant when d = 12? How fast is the radius changing at the instant when d = 16? When is the radius changing more rapidly, when d = 12 or when d = 16? A. when d = 12 B. when d = 16 C. Neither; the rate of change of the radius is constant