Need help on this problem please! Been stuck for half hour trying to figure it out but I can't get through a. A and B look to be similar but I don't know how to do them! Please help!
-----------------------------------
The resistance of blood flow, R, in a blood vessel is dependent on the length of the blood vessel, the radius of the blood vessel, and the viscosity of the blood. This relationship is given by R= 8Lη/π(r^4), where r is the radius, L is the length, and the positive value η is
the viscosity (the larger the value of η the more viscous the blood).
(a) Find the derivative of R with respect to r. Is the derivative positive or negative?
What is the physical interpretation of this? Does this result make sense?
(b) Find the derivative of R with respect to η. Is the derivative positive or negative?
What is the physical interpretation of this? Does this result make sense?
(c) As body temperature increases, the blood vessel will widen (i.e. radius increases)
and the blood becomes less viscous. Similarly as the body temperature lowers,the blood vessel will constrict and the blood will also become more viscous. We see then that the radius of the blood vessel and the viscosity of the blood are dependent on temperature. However, the length of the blood vessel essentially remains constant with temperature changes. Find the rate of change of resistance.
To find the derivative of a function, we can use the rules of differentiation. Let's go through each part of the problem step by step.
(a) To find the derivative of R with respect to r, we need to differentiate each term of the equation. Using the power rule and the chain rule, we get:
dR/dr = (8Lη/π) * (-4r^3)
Simplifying, we have:
dR/dr = -32Lηr^3/π
The derivative is negative, which means that as the radius increases, the resistance of blood flow decreases. This makes sense because as the blood vessel widens, there is more space for blood to flow, resulting in lower resistance.
(b) To find the derivative of R with respect to η, we treat L and r as constants since we are differentiating with respect to η. Therefore, the derivative of R with respect to η is:
dR/dη = (8L/π) * r^4
The derivative is positive, indicating that as the viscosity of the blood increases, the resistance of blood flow also increases. This makes sense because as the blood becomes more viscous, it encounters more resistance within the blood vessel.
(c) Since the length of the blood vessel essentially remains constant with temperature changes, we need to find the rate of change of resistance with respect to temperature. Here, the rate of change of resistance is the derivative of R with respect to temperature, which is not explicitly given in the equation. Without a specific equation relating temperature to the radius and viscosity, it is not possible to find the exact rate of change.