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
This is a word problem regarding differentiation and coughing.