Question

Q8.(10 points) a)When you cough,the radius of your trachea (windpipe) decreases,affecting the speed of the air in the trachea. If 0 r 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 2 0 S r ar r r ( ) ( ) = − , where a is positive constant. Find the radius r for which the speed of the air is greatest

Answer 1

`\bf{r=(2r_o)/(3)}` is greatest

Step-by-step explanation:

Given:

`S(r)=ar^2(r_(o)-r)`

`(ds)/(dr)=2ar(r_(o)-r)+ar^2(-1)`

`(ds)/(dr)=2ar_(o)-2ar^2-ar^2`

`(ds)/(dr)=2ar_(o)-3ar^2`

Substitute
`(ds)/(dr) = 0`, so

`2ar_(o)-3ar^2=0`

Then, get the common value from the equation.

`ra(2r_(o)-3r) = 0`

`\bf{r=0}\\r=(2r_(o))/(3)`

`((d^2s)/(dr^2))_(r=0)=2ar_(o)-6ar`

`((d^2s)/(dr^2))_(r=0)=2ar_(o) >0\;and, \\((d^2s)/(dr^2))_(r)=(2r_o)/(3) <0`

So, the speed of the air is greatest.

`\bf{r=(2r_o)/(3)}\;is\;greatest`