Answer
1.0 m.
Work Step by Step
The bottle is raised so that the fluid pressure is 78 torr higher than air pressure as it enters the vein. The pressure at the entrance to the needle is even higher than that. Use Poiseuille’s equation to calculate the required excess pressure. Equation 10–3c (page 266) tells us the height of the blood bottle.
$$Q=\frac{\pi R^4 (P_1-P_2)}{8 \eta_{blood} \mathcal{l}}$$
$$P_1=P_2+\frac{8 \eta_{blood} \mathcal{l}Q}{\pi R^4}=\rho_{blood}g\Delta h$$
$$\Delta h =\frac{1}{\rho_{blood}g }( P_2+\frac{8 \eta_{blood} \mathcal{l}Q}{\pi R^4}) $$
Putting in the pressure of 78 torr and using the other numbers given, the height is 1.0 m.