Final answer:
The gauge pressure needed to achieve a water speed of 12 m/s from the small end of a service hose can be calculated using the formula ∏P = ρgh, where ∏P is the change in pressure, ρ is the density of water, g is acceleration due to gravity, and h is the height difference.
Step-by-step explanation:
In order for water to emerge from the small end of a service hose with a speed of 12 m/s, the gauge pressure must be calculated based on the height difference between the small and large ends of the hose. Using the formula ∏P = ρgh, where ∏P is the change in pressure, ρ is the density of water, g is acceleration due to gravity, and h is the height difference, we can calculate the gauge pressure needed to achieve the desired speed.