When coughing, the windpipes contract to increase the velocity of air passing through the windpipe to help clear mucus. The velocity, U, at which the air flows through the windpipe depends on the radius, r of the windpipe. If R is the resting radius of the windpipe, then the velocity of air passing through the windpipe satisfies: v(r) = Ar²(R – r), where A is a constant dependent on the strength of the diaphram muscles. Find the derivative of the velocity function: (Note that your answers in the questions below may include A and R.) v'(r) = Find the value of r that maximizes the velocity of air. Imax = Determine the maximum velocity of the air flowing through the windpipe. v(rmax) =

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