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According to B. F. Visser, the velocity v of air in the trachea during a cough is related to the radius r of the trachea according to the following, where a is a constant and r0 is the radius of the trachea in a relaxed state. v = ar2(r0 − r) Find the radius r that produces the maximum velocity of air in the trachea during a cough.

Respuesta :

Answer:

[tex]r=\dfrac{2r_o}{3}[/tex]

Explanation:

Given that

v = ar²(ro − r)

v= a (ro .r² - r³)

For maximum velocity

[tex]\dfrac{dv}{dr}=0[/tex]

Let find the value of dv/dr

v= a (ro .r² - r³)

[tex]\dfrac{dv}{dr}=a .r_o.2 r - 3 a r^2[/tex]

[tex]\dfrac{dv}{dr}=0[/tex]

a .ro.2 r - 3 r² = 0

2 a ro = 3 r

[tex]r=\dfrac{2r_o}{3}[/tex]

So at the [tex]r=\dfrac{2r_o}{3}[/tex] velocity will be maximum.