# optimization and coughing - max-min word-problem

• Dec 28th 2012, 01:38 PM
zachd77
optimization and coughing - max-min word-problem
Got this problem on a test wrong, I misread it quite badly.

According to one model of coughing, the flow F (volume per unit time) of air through the windpipe during a cough is a function of the radius r of the windpipe, given by:

Attachment 26385 for Attachment 26386, where k is a constant and r0 is the normal (non-coughing) radius.

a) Find the value of r that maximizes the flow F.
b) According to the same model, the velocity v of air through the windpipe during a cough is given by:

Attachment 26387 for Attachment 26386. Find the value of r that maximizes the velocity v.

c) During a cough, the windpipe is constricted. According to parts (a) & (b), is that likely to assist or hinder the cough?

Lengthy problem. On my test I differentiated incorrectly, and that messed up my whole problem. How do I do this right?
• Dec 28th 2012, 02:00 PM
skeeter
Re: optimization and coughing - max-min word-problem
$\displaystyle F = k(r_0 \cdot r^4 - r^5)$

$\displaystyle \frac{dF}{dr} = k(4r_0 \cdot r^3 - 5r^4)$

$\displaystyle \frac{d^2F}{dr^2} = k(12r_0 \cdot r^2 - 20r^3) = 4kr^2(3r_0 - 5r)$

setting $\displaystyle \frac{dF}{dr} = 0$ ...

$\displaystyle 4r_0 \cdot r^3 - 5r^4 = 0$

$\displaystyle r^3(4r_0 - 5r) = 0$

$\displaystyle r = \frac{4r_0}{5}$ ... maximum $\displaystyle F$ since $\displaystyle \frac{d^2F}{dr^2} < 0$ for this value of $\displaystyle r$

$\displaystyle v = \frac{k}{\pi}(r_0 \cdot r^2 - r^3)$

$\displaystyle \frac{dv}{dr} = \frac{k}{\pi}(2r_0 \cdot r - 3r^2)$

$\displaystyle \frac{d^2v}{dr^2} = \frac{k}{\pi}(2r_0 - 6r)$

setting $\displaystyle \frac{dv}{dr} = 0$ ...

$\displaystyle 2r_0 \cdot r - 3r^2 = 0$

$\displaystyle r(2r_0 - 3r) = 0$

$\displaystyle r = \frac{2r_0}{3}$ ... maximum for $\displaystyle v$ since $\displaystyle \frac{d^2v}{dr^2} < 0$ for this value of $\displaystyle r$
• Dec 28th 2012, 02:50 PM
zachd77
Re: optimization and coughing - max-min word-problem
So, I'd say the that if the windpipe is constricted, the cough would most likely be hindered. But how does that relate to the problem (how do the answers influence the last part)?
• Dec 28th 2012, 03:17 PM
skeeter
Re: optimization and coughing - max-min word-problem
Since flow and velocity are a maximum when the windpipe is restricted, I would say it helps, rather than hinders the cough.
• Dec 28th 2012, 04:44 PM
zachd77
Re: optimization and coughing - max-min word-problem
now I see... have to watch my time from now on.