Final answer:
To determine the gauge pressure required for water to flow through a pipe with a tapered shape and an elevation difference, we can use Bernoulli's equation.
Step-by-step explanation:
In this question, we are given a pipe that tapers, with the large end having a diameter twice as large as the small end. We are asked to determine the gauge pressure required for water to emerge from the small end with a speed of 12 m/s, when the small end is elevated 8 m above the large end of the pipe.
To solve this, we need to consider the principle of Bernoulli's equation, which states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume of fluid is constant along a streamline. By applying Bernoulli's equation and solving for the gauge pressure, we can find the answer.
The specific equation we can use is: P + (1/2)ρv^2 + ρgh = constant, where P is the pressure, ρ is the density of water, v is the velocity of water, g is the acceleration due to gravity, and h is the vertical height difference.