Question

A water tower is a familiar sight in many towns. The purpose of such a tower is to provide storage capacity and to provide sufficient pressure in the pipes that deliver the water to customers. The drawing shows a spherical reservoir that contains 7.02 x 105 kg of water when full. The reservoir is vented to the atmosphere at the top. For a full reservoir, find the gauge pressure that the water has at the faucet in (a) house A and (b) house B. Ignore the diameter of the delivery pipes.

Answer 1

Note: The drawing referred to in the question is attached below

Answer:

gauge pressure that the water has at the faucet in house A = 254878.4 Pa

gauge pressure that the water has at the faucet in house B = 186278.4 Pa

Explanation:

Mass of water, $m = 7.02 * 10^5 kg$

Calculate the volume of water in the reservoir:

$V = m/\rho$

Where $\rho$ = density of water = 10³ kg/$m^3$

$V = \frac{7.02 * 10^5}{10^3} \\V = 702 m^3$

Since the reservoir is spherical in shape, the volume of a sphere is given by the equation:

$V = \frac{4}{3} \pi r^{3} \\702 = \frac{4}{3} \pi * r^{3} \\r = 5.504 m$

By observing the diagram shown, the height of the tower to house A:

h = 2r + 15 = 2(5.504) + 15 = 26.008 m

The gauge pressure in house A can be given by the formula:

$P_A = \rho gh\\P_A = 1000 * 9.8 *26.008\\P_A = 254878.4 Pa$

By observing the diagram shown, the height of the tower to house B:

h = 2r + 15 - 7 = 2(5.504) + 15 - 7

h= 19.008 m

$P_B = \rho gh\\P_A = 1000 * 9.8 *19.008\\P_B = 186278.4 Pa$