optimization

[mlane]

I have a few problems on optimization that I don't have a clue what to do with, maybe someone can direct me on a couple.

problem 1.
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 windipe, then for r<R, the speed is given by:
v = a(R-r)r^2 where a is a positive constant.
What value of r maximizes the speed.

Problem 2

For some positive constant C, a patient's temperature change, T, due to a dose, D of a drug is given by
T = (C/2-D/3)D^2

a) what dosage maximizes the temperature change?
b) The sensitivity of the body to the drug is defined as dT/dD. What dosage maximizes sensitivity?[/tex]

 

[Unco]

G'day mlane,

problem 1.
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 is given by:
v = a(R-r)r^2 where a is a positive constant.
What value of r maximizes the speed.

Which part are you having trouble with:
1. Differentiate v with respect to r. You should expand the parentheses first. R and a are just constants.

2. Set v' (the derivative you just found) to zero.

3. Factorise.

4. Solve for r, noting that the radius of the windpipe is not going to be zero, so the other other value for r will maximise the speed.

If you want to show that value is indeed a maximum, you would want to find the second derivaitve and test its sign at this r value.

 

[mlane]

I think I am messing up the derrivative

I have V = (aR-ar)r^2
then v =aRr^2-ar^3
derrivative? 1=(2ar-3ar^2)dr/dv
next I am not sure? I think I am messing up my algebra after this?
Does the derrivative look right?

 

[Unco]

What happened to R? Apart from that your derivative is fine. Note that you want to have dv/dr on one side by itself so you can set it zero.

You'll have to show your work if we are to help you check your algebra .

 

[mlane]

I got it figured out and thanks for your help, I think I mistyped part of it. I didn't know that the dv/dr could be set to zero and that was messing with my thinking. I think initially, I was taking the derivative with respect to V instead of r.