Coughing forces the trachea (windpipe) to contract, which in turn affects the velocity of the air through the trachea. The velocity of the air during coughing can be modeled by v = k (R - r) r2, 0 ≤ r < R, where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius r will produce the maximum air velocity?

Question

Mathematics

Ali Hancock

11 November, 13:40

Coughing forces the trachea (windpipe) to contract, which in turn affects the velocity of the air through the trachea. The velocity of the air during coughing can be modeled by v = k (R - r) r2, 0 ≤ r < R, where k is a constant, R is the normal radius of the trachea, and r is the radius during coughing. What radius r will produce the maximum air velocity?

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Answers (1)

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Julia Frank · Answer 1

To find the maximum velocity you have to differentiate the function respect to r.

v = k (R - r) r^2 = kRr^2 - krr^2 = kRr^2 - kr^3

k and R are constants.

=> v' = 2kRr - 3kr^2 = kr (2R - 3r)

maximum velocity = > v' = 0 = > kr (2R - 3r) = 0

r = 0 or 2R - 3r = 0 = > r = 2R / 3

Answer: r = 2R / 3