- #1

Razael

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## Homework Statement

When you cough, your windpipe contracts. The speed, v, with which air comes out depends on the radius, r, of your windpipe. If R is the normal (rest) radius of your windpipe, then for r <_ R, the speed if given by v = a(R-r)r

^{2}where

*a*is a positive constant. What value of r maximizes v? What is the maximum speed? Show all work. Use the procedure for finding global extrema to justify that you have found the global maximum velocity.

## Homework Equations

v = a(R-r)r

^{2}

## The Attempt at a Solution

Derivative is 2ar(R-r) - ar

^{2}(R is considered a constant, right?)

= 2arR - 2ar

^{2}-ar

^{2}

= 2arR - 3ar

^{2}= 0

2arR = 3ar

^{2}

2aR = 3ar

r = (2/3)R

Not too confident in that answer. For maximum speed I just plugged it into the original equation:

v = a((2/3)R)

^{2}((1/3)R)

Any help is appreciated, whether it be by pointing out a mistake or helping with the last part (justify answer).

**Edit**The interval would be 0 < r <_ R, right?

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