# Strange angle differentiation

• May 31st 2008, 07:53 PM
Blackstar347
Strange angle differentiation
Ok first time being here but I could use some help. The problem is this.

-There are 2 blood vessels seperated by angle X
-These two blood vessels have radii of R and r respectively
-The length "a" is the length of the primary blood vessel while the height "b" is the length from the end of the primary blood vessel to the branching vessel.
-The resistance to blood flow is as follows:
T= ((a-bcot(x))/R^4)-((bcsc(x))/r^4
-R,r,a, and b are all constants.
-Which angle "x" would provide the least resistance of blood?

I worked out the derivative to look like
T'=((bcsc(x)^2/R^4)+((bcsc(x)cot(x)/r^4)

But I have no idea how to get a minimum of resistance since the only angles to make this 0 or undefined are angles such as 0 90 or 180 which are not the answers. Have I done something wrong in my derivative? Is there a specific answer?
• June 1st 2008, 04:51 AM
mr fantastic
Quote:

Originally Posted by Blackstar347
Ok first time being here but I could use some help. The problem is this.

-There are 2 blood vessels seperated by angle X
-These two blood vessels have radii of R and r respectively
-The length "a" is the length of the primary blood vessel while the height "b" is the length from the end of the primary blood vessel to the branching vessel.
-The resistance to blood flow is as follows:
T= ((a-bcot(x))/R^4)+((bcsc(x))/r^4
-R,r,a, and b are all constants.
-Which angle "x" would provide the least resistance of blood?

I worked out the derivative to look like
T'=((bcsc(x)^2/R^4)-((bcsc(x)cot(x)/r^4)

But I have no idea how to get a minimum of resistance since the only angles to make this 0 or undefined are angles such as 0 90 or 180 which are not the answers. Have I done something wrong in my derivative? Is there a specific answer?

First of all, I'm pretty sure there's meant to be a plus between the two terms in T since the first and second terms represent resistance to blood flow in the primary and branching blood vessel respectively. I've made the necessary corrections in red.

The derivative can be re-written as $\frac{dT}{dx} = \frac{b}{R^4 \sin^2 x} - \frac{b \cos x}{r^4 \sin^2 x} = \frac{b(r^4 - R^4 \cos x)}{R^4 r^4 \sin^2 x}$.

$\frac{dT}{dx} = 0 \Rightarrow r^4 - R^4 \cos x = 0 \Rightarrow \cos x = \left(\frac{r}{R}\right)^4$.

There will be a solution since r < R (why?).

Now you must test the nature of the solution. Given the model, you're only interested in solutions lying in the domain $\left(\alpha, \, \frac{\pi}{2} \right)$ where $\alpha > 0$ will depend on the value of a and b (I'll let you work out the relationship).
• June 1st 2008, 09:34 AM
Blackstar347
Well the branching blood vessel has a smaller radius than the main blood vessel which is why r<R...hmm I feel like I'm missing a value somewhere. But thanks for the tips the derivative there seems alot easier to understand.