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).
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.
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
patience, patience, you had it :)
I am so appreciative of your help, but what happened to the b?
b is a constant so
when
0 = b * (something - b * (something else)
b cancels
To find the value of θ that minimizes the total resistance, we can take the derivative of the resistance formula with respect to θ and set it equal to zero.
First, let's recall the derivatives of some trigonometric functions:
d/dθ (cotθ) = -csc^2θ
d/dθ (cscθ) = -cscθcotθ
Now, let's find the derivative of the resistance formula:
d/dθ (T) = d/dθ [(a - b·cotθ)/4^4 + (b·cscθ)/2^4]
Using the chain rule, we can differentiate each term separately:
d/dθ [(a - b·cotθ)/4^4] = -b/4^4 · d/dθ (cotθ) = -b/4^4 · (-csc^2θ) = b·csc^2θ/4^4
d/dθ [(b·cscθ)/2^4] = b/2^4 · d/dθ (cscθ) = b/2^4 · (-cscθ·cotθ) = -b·cscθcotθ/2^4
Combining both derivatives, we get:
d/dθ (T) = b·csc^2θ/4^4 - b·cscθcotθ/2^4
Now, we set this derivative equal to zero to find the critical points:
b·csc^2θ/4^4 - b·cscθcotθ/2^4 = 0
Divide both sides by b to simplify the equation:
csc^2θ/4^4 - cscθcotθ/2^4 = 0
Now, we can multiply through by 2^4 to get rid of the denominators:
16·csc^2θ - 4·cscθcotθ = 0
Divide through by 4 to further simplify:
4·csc^2θ - cscθcotθ = 0
Now, substitute the identity cscθ = 1/sinθ and simplify:
4/sin^2θ - 1/sinθ·cotθ = 0
Multiply through by sin^2θ to eliminate the denominators:
4 - sinθ·cotθ = 0
Rearrange the equation by multiplying through by sinθ:
4sinθ - sinθ·cotθ·sinθ = 0
Using the identity cotθ = cosθ/sinθ:
4sinθ - sinθ·cosθ/sinθ·sinθ = 0
Simplify:
4sinθ - cosθ/sinθ = 0
Multiply through by sinθ to eliminate the denominators:
4sin^2θ - cosθ = 0
Using the identity sin^2θ + cos^2θ = 1:
4(1 - cos^2θ) - cosθ = 0
Simplify:
4 - 4cos^2θ - cosθ = 0
Rearrange the equation:
4cos^2θ + cosθ - 4 = 0
Now, we have a quadratic equation in terms of cosθ. We can solve for cosθ using the quadratic formula:
cosθ = (-1 ± √(1^2 - 4·4·(-4))) / (2·4)
cosθ = (-1 ± √(1 + 64)) / 8
cosθ = (-1 ± √65) / 8
Now, to find the value of θ that minimizes the total resistance, we need to determine which of the two solutions for cosθ satisfies the given constraints.
Since the radius of the small blood vessel is 2mm and the radius of the large blood vessel is 4mm, we can infer that θ must be an acute angle (θ < π/2).
Therefore, we choose the positive solution for cosθ:
cosθ = (1 - √65) / 8
Hence, cosθ = 1/16 minimizes the total resistance to blood flow according to Poiseuille's Law.