I have the question :A small blood vessel of radius 2mm branches off at an angle(theta) from a larger blood vessel of radius 4mm. According to Poiseuille's Law the total resistance to the blood flow is proportional to T=(a-bcot(theta)/4^4)+(bcsc(theta)/2^4) Show that the total resistance is minimized when cos(theta)=(1/16).

Question

I think i need to find the derivative, which should be (bcsc^2(theta)/4^4)-(bcsc(theta)cot(theta)/2^4).
I have no clue if that is even correct, or what to do to solve this problem.

Answer 1

T=(a-bcot(theta)/4^4)+(bcsc(theta)/2^4)

dT/dtheta =b times
-1 (-csc^2)/4^4
+1 (-csc ctn )/2^4

set the derivative to zero for max or min

csc^2/2^8 = csc ctn /2^4
csc /ctn = 2^8/2^4 = 16

1/sin / cos/sin = 16
1/cos = 16
cos = 1/16

Answer 2

patience, patience, you had it :)

Answer 3

I am so appreciative of your help, but what happened to the b?

Answer 4

b is a constant so
when
0 = b * (something - b * (something else)
b cancels