the flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the outside. the speed of the blood v, in millimeters per second (mm/sec) is given by the following formula, where r is the radius of the blood vessel, r is the distance of the blood from the center of the vessel, and p, l, and v are physical constants related to pressure, length, and viscosity of the blood vessel, respectively. assume that r is a constant as well as p, l, and v. complete parts (a) and (b) below.

v = \frac{p}{4l\
u}(r^{2}-r^{2})

a) find the rate of change \frac{dv}{dt} in terms of r (in mm) and \frac{dr}{dt} (in mm/sec) when l = 80 mm, p = 450 pa, and v = 0.003 pa·sec. select the correct answer below and fill in the answer box to complete your choice.

a. \frac{dv}{dt}=187.5r·\frac{dr}{dt}

b. \frac{dv}{dt}=·\frac{dr}{dt}

b) when shivering occurs in cold air, a person with a history of heart trouble can develop angina (chest pains) due to contracting blood vessels. to counteract this, they may take a nitroglycerin tablet, which dilates blood vessels. suppose that after a nitroglycerin tablet is taken, a blood vessel dilates at a rate of \frac{dr}{dt}=0.00013 mm/sec at a place in the blood vessel where the radius r = 0.598 mm. find the rate of change \frac{dv}{dt}

\frac{dv}{dt}=\square

(round to four decimal places as needed.)

Question

the flow of blood in a blood vessel is faster toward the center of the vessel and slower toward the outside. the speed of the blood v, in millimeters per second (mm/sec) is given by the following formula, where r is the radius of the blood vessel, r is the distance of the blood from the center of the vessel, and p, l, and v are physical constants related to pressure, length, and viscosity of the blood vessel, respectively. assume that r is a constant as well as p, l, and v. complete parts (a) and (b) below.

v = \frac{p}{4l\
u}(r^{2}-r^{2})

a) find the rate of change \frac{dv}{dt} in terms of r (in mm) and \frac{dr}{dt} (in mm/sec) when l = 80 mm, p = 450 pa, and v = 0.003 pa·sec. select the correct answer below and fill in the answer box to complete your choice.

a. \frac{dv}{dt}=187.5r·\frac{dr}{dt}

b. \frac{dv}{dt}=·\frac{dr}{dt}

b) when shivering occurs in cold air, a person with a history of heart trouble can develop angina (chest pains) due to contracting blood vessels. to counteract this, they may take a nitroglycerin tablet, which dilates blood vessels. suppose that after a nitroglycerin tablet is taken, a blood vessel dilates at a rate of \frac{dr}{dt}=0.00013 mm/sec at a place in the blood vessel where the radius r = 0.598 mm. find the rate of change \frac{dv}{dt}

\frac{dv}{dt}=\square

(round to four decimal places as needed.)

Accepted Answer

# Explanation:
## Step1: Substitute the given values into the formula for $V$
Given $V=\frac{p}{4L
u}(R^{2}-r^{2})$, substitute $L = 80$ mm, $p = 450$ Pa, and $
u=0.003$ Pa·sec.
$$
\begin{align*}
\frac{p}{4L
u}&=\frac{450}{4\times80\times0.003}\
&=\frac{450}{0.96}\
& = 468.75
\end{align*}
$$
So $V = 468.75(R^{2}-r^{2})$. Since $r$ is a constant, when differentiating with respect to $t$, $\frac{dV}{dt}=468.75\times2R\frac{dR}{dt}$ (using the chain - rule $\frac{d}{dt}(u^{2}) = 2u\frac{du}{dt}$ where $u = R$). Then $\frac{dV}{dt}=937.5R\frac{dR}{dt}$

## Step2: Calculate $\frac{dV}{dt}$ for part (b)
We know from part (a) that $\frac{dV}{dt}=937.5R\frac{dR}{dt}$. Given $R = 0.976$ mm and $\frac{dR}{dt}=0.00013$ mm/sec.
$$
\begin{align*}
\frac{dV}{dt}&=937.5\times0.976\times0.00013\
&=937.5\times0.00012688\
&=0.119
\end{align*}
$$

# Answer:
a) $\frac{dV}{dt}=937.5R\frac{dR}{dt}$
b) $\frac{dV}{dt}=0.119$ mm³/sec

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QUESTION IMAGE

Question

Explanation:

Step1: Substitute the given values into the formula for \(V\)

Step2: Calculate \(\frac{dV}{dt}\) for part (b)

Answer: