Physiologists define the flow F of air in the trachea by the formula F = SA, where S is the speed of the air and A is the area of a cross section of the trachea.

a. Assume the trachea has a circular cross section with radius r. Use the formula for the speed of air in the trachea during a cough given in Example 3.4.3 to express air flow F in terms of r.

b. Find the radius r for which the flow is greatest.

Data from Example 3.4.3 

Finding Maximum Air Speed During a Cough When you cough, the radius of your trachea (windpipe) decreases, affecting the speed of the air in the trachea. If r₀ is the normal radius of the trachea, the relationship between the speed S of the air and the radius r of the trachea during a cough is given by a function of the form S(r) = ar²(r₀ − r), where a is a positive constant.* Find the radius r for which the speed of the air is greatest.

Transcribed Image Text:

10= A = πr²