Using the property table,
The density of the blood is given as,
\(\rho = 1050\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\)
The viscosity of the blood is given as,
\(\eta = 4.0 \times {10^{ - 3}}\;{\rm{Pa}} \cdot {\rm{s}}\)
Applying the Poiseuille’s law, the expression for the pressure drops between the inlet and exit of the needle is given as,
\(\begin{array}{c}Q = \frac{{\pi {D^4}\Delta P}}{{128\eta L}}\\3.33 \times {10^{ - 8}}\;{{\rm{m}}^3}{\rm{/s}} = \frac{{\pi \times {{\left( {0.0008\;{\rm{m}}} \right)}^4} \times \Delta P}}{{128 \times 4.0 \times {{10}^{ - 3}}\;{\rm{Pa}} \cdot {\rm{s}} \times 0.025\;{\rm{m}}}}\\\Delta P = 331.24\;{\rm{Pa}}\\{P_{exit}} - {P_{blood}} = 331.24\;{\rm{Pa}}\end{array}\)
Substituting the value in the above equation,
\(\begin{array}{c}{P_{exit}} = 331.24\;{\rm{Pa}} + {P_{blood}}\\{P_{exit}} = 331.24\;{\rm{Pa}} + 10397.4\;{\rm{Pa}} + {P_{atm}}\\{P_{exit}} = 10728.64\;{\rm{Pa}} + {P_{atm}}\end{array}\)
Using hydrostatic law, the change in pressure between the top of the bottle and the exit of the needle is given as,
\(\begin{array}{c}{P_{top}} + \rho gh = {P_{exit}}\\{P_{atm}} + 1050\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}} \times 9.81\;{\rm{m/}}{{\rm{s}}^2} \times h = 10728.64\;{\rm{Pa}} + {P_{atm}}\\10300.5\;{\rm{kg/}}{{\rm{m}}^2} \cdot {{\rm{s}}^2} \times \frac{{1\;{\rm{N}}}}{{1\;{\rm{kg}} \cdot {\rm{m/}}{{\rm{s}}^2}}} \times h = 10728.64\;{\rm{Pa}} \times \frac{{1\;{\rm{N/}}{{\rm{m}}^2}}}{{1\;{\rm{Pa}}}}\\h = 1.04\;{\rm{m}}\end{array}\)
Thus, the bottle should be placed \(1.04\;{\rm{m}}\) above the needle.