When a person coughs his or her trachea windpipe contracts...

When a person coughs, his or her trachea (windpipe) contracts, allowing air to be expelled at a maximum velocity. It can be shown that during a cough, the velocity v of the airflow is given by the function v = f(r) = kr^2(R - r), where r is the trachea's radius during a cough, R is the trachea's normal radius (both measured in centimeters), and k is a positive constant. Find the radius r for which the velocity of the airflow is the greatest (that is, find the absolute maximum value of v on the interval [0, R]). Your answer will be in terms of R.

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Transcript

00:01 Hi there! so for this problem, we are given the following expression for this speed.

00:08 So that will be k, which is a constant, this times r to the square, which is the trakeous radius.

00:21 And this times the normal trakeous radius, this minus the radius, rt, so with that said, what we need to find is the radius r, for which the velocity of the airflow is the greatest.

00:47 So what we need to find in this case is the maximum for this function.

00:51 So we take the derivative of this with respect to the radius art, and then we set that equal to zero.

01:00 So from there, solving for the radius, we'll give us the radius that, that maximize the velocity.

01:10 So with that said, let us just first derivate this expression.

01:15 So the derivative of this velocity with respect to time, as you can see, we have a product in here between two functions.

01:24 So what we can do is to first derivate by leaving one expression as a constant.

01:31 So for example, we will have 2 times k times the radios.

01:35 This times the radius minus r.

01:38 So we leave as a constant this term in here.

01:42 Now we keep this term as a constant and then thereby this other turn.

01:48 So that will be just simply.

01:55 Let me just see here...

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