Question

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.

Answer 1

`r=(2r_o)/(3)`

Step-by-step explanation:

Given that

v = ar²(ro − r)

v= a (ro .r² - r³)

For maximum velocity

`(dv)/(dr)=0`

Let find the value of dv/dr

v= a (ro .r² - r³)

`(dv)/(dr)=a .r_o.2 r - 3 a r^2`

`(dv)/(dr)=0`

a .ro.2 r - 3 r² = 0

2 a ro = 3 r

`r=(2r_o)/(3)`

So at the
`r=(2r_o)/(3)` velocity will be maximum.