optimization and coughing - max-min word-problem

**zachd77** · December 28th 2012, 01:38 PM

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:

optimization and coughing - max-min word-problem-codecogseqn-7.gif

for

optimization and coughing - max-min word-problem-codecogseqn-8.gif

, 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:

for . 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?

**skeeter** · December 28th 2012, 02:00 PM

$F = k(r_0 \cdot r^4 - r^5)$

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

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

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

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

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

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

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

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

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

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

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

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

$r = \frac{2r_0}{3}$ ... maximum for $v$ since $\frac{d^2v}{dr^2} < 0$ for this value of $r$

**zachd77** · December 28th 2012, 02:50 PM

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)?

**skeeter** · December 28th 2012, 03:17 PM

Since flow and velocity are a maximum when the windpipe is restricted, I would say it helps, rather than hinders the cough.

**zachd77** · December 28th 2012, 04:44 PM

now I see... have to watch my time from now on.

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