From Bernoulli's principle:

$\overline{){{\mathbf{P}}}_{{\mathbf{1}}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{\rho}}}_{{\mathbf{1}}}{{{\mathbf{v}}}_{{\mathbf{1}}}}^{{\mathbf{2}}}{\mathbf{+}}{{\mathbf{\rho}}}_{{\mathbf{1}}}{{\mathbf{h}}}_{{\mathbf{1}}}{\mathbf{g}}{\mathbf{=}}{{\mathbf{P}}}_{{\mathbf{2}}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{\rho}}}_{{\mathbf{2}}}{{\mathbf{v}}_{\mathbf{2}}}^{{\mathbf{2}}}{\mathbf{+}}{{\mathbf{\rho}}}_{{\mathbf{2}}}{{\mathbf{h}}}_{{\mathbf{2}}}{\mathbf{g}}{\mathbf{=}}{\mathbf{c}}{\mathbf{o}}{\mathbf{n}}{\mathbf{s}}{\mathbf{t}}{\mathbf{a}}{\mathbf{n}}{\mathbf{t}}}$

**(a)**

Since it has the same velocity, the equation is reduced to:

${{\mathbf{P}}}_{{\mathbf{1}}}{\mathbf{+}}{{\mathbf{\rho}}}_{{\mathbf{1}}}{{\mathbf{h}}}_{{\mathbf{1}}}{\mathbf{g}}{\mathbf{=}}{{\mathbf{P}}}_{{\mathbf{2}}}{\mathbf{+}}{{\mathbf{\rho}}}_{{\mathbf{2}}}{{\mathbf{h}}}_{{\mathbf{2}}}{\mathbf{g}}$

P_{2} is obtained as:

$\begin{array}{rcl}{{\mathbf{P}}}_{{\mathbf{2}}}& \mathbf{=}& {{\mathbf{P}}}_{{\mathbf{1}}}{\mathbf{+}}{{\mathbf{\rho}}}_{{\mathbf{1}}}{{\mathbf{h}}}_{{\mathbf{1}}}{\mathbf{g}}{\mathbf{-}}{{\mathbf{\rho}}}_{{\mathbf{2}}}{{\mathbf{h}}}_{{\mathbf{2}}}{\mathbf{g}}\\ & \mathbf{=}& {{\mathbf{P}}}_{{\mathbf{1}}}{\mathbf{+}}{\mathbf{\rho g}}{\mathbf{(}}{\mathbf{\u2206}}{\mathbf{h}}{\mathbf{)}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{00}\mathbf{\times}{\mathbf{10}}^{\mathbf{5}}\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{\times}{\mathbf{10}}^{\mathbf{3}}\mathbf{)}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{50}\mathbf{)}\end{array}$

A sump pump (used to drain water from the basement of houses built below the water table) is draining flooded basement at a rate of 0.750 L/s, with an output pressure of 3.00x10^{5} N/m^{2}.

(a) The water enters a hose with a 3.00-cm inside diameter and rises 2.50 m above thew pump. What is the pressure at this point?

(b) the hose goes over the foundation wall, losing 0.500 m in height, and widens to 4.00 cm in diameter. what is the pressure now? You may neglect frictional losses in both parts of the problem.

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