# Question here. on max

• Dec 21st 2006, 11:24 PM
Dave J
Question here. on max
The question is.

The speed of air escaping through your trachea when you cough is a function of the force due to chest contraction and the size of the trachea. It can be modelled by the equation
S = F(1+0.9r^2-r^3) where F is the constant force of youre chest contraction, r is the radius of your trachiea in centimeters, S is the speed of the airflow.

Determine the max speed of airflow.

I know to find the max speed I need to take the derivative and find critical numbers, but how do I do that when there is the F and the r??
• Dec 22nd 2006, 12:15 AM
CaptainBlack
Quote:

Originally Posted by Dave J
The question is.

The speed of air escaping through your trachea when you cough is a function of the force due to chest contraction and the size of the trachea. It can be modelled by the equation
S = F(1+0.9r^2-r^3) where F is the constant force of youre chest contraction, r is the radius of your trachiea in centimeters, S is the speed of the airflow.

Determine the max speed of airflow.

I know to find the max speed I need to take the derivative and find critical numbers, but how do I do that when there is the F and the r??

F is a constant, you are required to find the maximum of S as r varies.

dS/dr=F(1.8r-3r^2)

so the stationary points correspond to the roots of:

1.8r - 3 r^2=0,

which are r=0, and r=0.6. Now d^2S/dr^2 is positive at r=0, so this is
a local minimum, and d^2S/dr^2 is negative at r=0.6 so this is a local
maximum, and S(0.6)=1.108 F.

This is also a global maximum, as r must be greater than 0.

RonL