Chapter 2.2, Problem 62E

Coughing velocity. A person coughs when a foreign object is in the windpipe. The velocity of the cough depends on the size of the object. Suppose a person has a windpipe with a 20-mm radius. If a foreign object has a radius r, in millimeters, then the velocity V, in millimeters per second, needed to remove the object by a cough is given by

$V\left(r\right)=k\left(20r^2-r^3\right),0\le r\le 20,$

where k is some positive constant. For what size object is the maximum velocity required to remove the object?

$\text{Radius }=\frac{40}{3},\text{ or }13\frac{1}{3}\text{ mm}$