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 $(\mathrm{mm} / \mathrm{sec})$, is given by the following formula, where $\mathrm{R}$ 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 vessels, respectively. Assume that $r$ is a constant as well as $p, L$, and $v$. Complete parts (a) and (b) below. \[ V=\frac{p}{4 L v}\left(R^{2}-r^{2}\right) \] a) Find the rate of change $\frac{d V}{d t}$ in terms of $R$ and $\frac{d R}{d t}$ when $L=75 \mathrm{~mm}, p=400$, and $v=0.003$. Select the correct answer below and fill in the answer box to complete your choice. A. $\frac{\mathrm{dV}}{\mathrm{dt}}=\square \cdot \frac{\mathrm{dR}}{\mathrm{dt}}$ B. $\frac{\mathrm{dV}}{\mathrm{dt}}=\square \cdot \frac{\mathrm{dr}}{\mathrm{dt}}$
Solution
Step 1 ：The given formula for the speed of the blood is \(V = \frac{p}{4Lv}(R^2 - r^2)\).
Step 2 ：We are asked to find the rate of change of the speed of the blood, \(\frac{dV}{dt}\), in terms of \(R\) and \(\frac{dR}{dt}\).
Step 3 ：Since \(r\) is a constant, it will not affect the rate of change of \(V\) with respect to time.
Step 4 ：We need to find the derivative of \(V\) with respect to \(R\), and then multiply it by \(\frac{dR}{dt}\), according to the chain rule of differentiation.
Step 5 ：The derivative of \(V\) with respect to \(R\) is \(888.888888888889R\).
Step 6 ：For every unit increase in \(R\), the speed of the blood, \(V\), increases by \(888.888888888889R\) units.
Step 7 ：We are asked to find the rate of change of \(V\) with respect to time, \(t\), not \(R\). According to the chain rule, we need to multiply this derivative by \(\frac{dR}{dt}\) to get \(\frac{dV}{dt}\).
Step 8 ：Final Answer: \(\boxed{\frac{dV}{dt} = 888.888888888889R \cdot \frac{dR}{dt}}\)
