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Answer :
The pressure required at the nozzle inlet for the decorative fountain, we can use the principle of conservation of energy. the pressure at the inlet to be psi.
1. First, let's convert the inlet and outlet diameters to feet. The inlet diameter is 4.0 inches, which is equivalent to 4.0/12 = 0.33 feet. The outlet diameter is 2.0 inches, equivalent to 2.0/12 = 0.17 feet.
2. Since the water shoots straight up, the velocity at the inlet is zero. Therefore, the initial kinetic energy is also zero.
3. The water reaches a height of 59 feet, which means that the final potential energy is mgh, where m is the mass of the water, g is the acceleration due to gravity (32.2 ft/s^2), and h is the height (59 ft).
4. The Bernoulli's equation states that the sum of pressure energy, kinetic energy, and potential energy remains constant. In this case, the kinetic energy is zero, so the equation becomes:
Pressure at the inlet + density of water × gravity × height at the inlet = pressure at the outlet + density of water × gravity × height at the outlet.
5. Rearranging the equation, we get:
Pressure at the inlet = pressure at the outlet + density of water × gravity × (height at the outlet - height at the inlet).
6. The density of water is approximately 62.4 lb/ft^3.
7. Substituting the values, we have:
Pressure at the inlet = pressure at the outlet + (62.4 lb/ft^3) × (32.2 ft/s^2) × (59 ft - 0 ft).
8. The pressure at the outlet is atmospheric pressure, which is about 14.7 psi.
9. Plugging in the values, we find:
Pressure at the inlet = 14.7 psi + (62.4 lb/ft^3) × (32.2 ft/s^2) × (59 ft).
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