Final answer:
The gauge pressure needed for water to exit a tapered pipe at a speed of 12 m/s, with the smaller end 8 m higher than the larger end, involves Bernoulli's equation considering kinetic and potential energy changes along with the pipe's diameter changes.
Step-by-step explanation:
To determine the gauge pressure for water to emerge from the small end of the pipe at a certain speed, when the diameter is twice as large at one end, and considering the elevation difference, we can apply Bernoulli's equation. Here we take into account the kinetic energy from the speed at which water emerges, the potential energy difference due to the elevation, and the pressure difference.
The gauge pressure can be found by considering the pressure-energy conversion due to change in elevation (gravitational potential energy), the change in kinetic energy (resulting from the water speed), and the continuity equation (since the diameter changes).
The specifics of calculating gauge pressure would typically involve the use of the equation derived from Bernoulli's principle, ensuring units are consistent to obtain an answer in Pascals (Pa).