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Differentiation and air velocity

Please show how to solve the problem to the specified answer. Please offer as much explanation as possible.

Coughing forces the trachea to contract, which affects the velocity v of the air passing through the trachea. Suppose the velocity of the air during coughing is

v = k(R - r)r^2, 0 is less than or equal to r which is less than R where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius will produce the maximum air velocity?
Answer: r = 2R/3

Solution Summary

This is a word problem regarding differentiation and coughing.

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