The number of times a new pop song has been downloaded $t$ weeks after its initial release is given by $f(t)=\frac{3,200,000}{1+300 e^{-0.85 t}}$. a) After how many weeks is the rate of change of the number of downloads maximized? b) What is the rate of change of the number of downloads at the time found in part (a)? c) How many times has the song been downloaded at the time found in part (a)?

Step 1 ：The rate of change of the number of downloads is given by the derivative of the function \(f(t)\). To find the time at which this rate is maximized, we need to find the maximum of the derivative function. This can be done by setting the second derivative of the function to zero and solving for \(t\).

Step 2 ：The derivative of the function \(f(t)\) is \(f'(t) = \frac{816000000.0 e^{-0.85t}}{(1 + 300 e^{-0.85t})^2}\) and the second derivative is \(f''(t) = -\frac{693600000.0 e^{-0.85t}}{(1 + 300 e^{-0.85t})^2} + \frac{416160000000.0 e^{-1.7t}}{(1 + 300 e^{-0.85t})^3}\).

Step 3 ：We set the second derivative equal to zero and solve for \(t\) to find the critical points. The critical points include complex numbers, which do not make sense in the context of this problem. We are only interested in the real number solutions.

Step 4 ：The real number solution is approximately 6.71. This is the time in weeks at which the rate of change of the number of downloads is maximized.

Step 5 ：Final Answer: The number of weeks after which the rate of change of the number of downloads is maximized is approximately \(\boxed{6.71}\).