Final answer:
Using the conservation of mass and the equation A1 * v1 = A2 * v2, we solve for the diameter of the opening in the pool toy. For the pressure in a lake, it is known that every 10 meters of depth adds roughly one atmosphere of pressure, so at 10 meters deep, the pressure is twice atmospheric pressure.
Step-by-step explanation:
To find the diameter of the opening through which water emerges in this popular pool toy, we can apply the principle of conservation of mass. Since the toy is a closed system, the mass flow rate of water through it must remain constant. The mass flow rate can be calculated by multiplying the cross-sectional area of the flow by the flow velocity. Using the formula A1 * v1 = A2 * v2 where A1 and A2 are the cross-sectional areas of the initial barrel and the opening, and v1 and v2 are the corresponding flow velocities, we can solve for the diameter of the opening (A = π * d^2/4).
As given, the water is compressed with a velocity of 1.2 m/s through a barrel with a diameter of 3.0 cm, and emerges at a velocity of 15 m/s. We calculate A1 to be π * (0.03 m)^2/4 and A2 to be A1 * v1 / v2. Thus, A2 = π * 0.0009 m^2/4 * (1.2 m/s / 15 m/s). Solving for the diameter gives us d = 2 * √(A2/π), which allows us to determine the diameter of the opening.
For the second part about pressure in a lake, pressure increases with depth due to the weight of the water above. Every 10 meters of water depth adds approximately one atmosphere (atm) of pressure. Therefore, to reach a pressure twice that of atmospheric pressure, we would go to a depth where the total pressure equals two atmospheres: one atmosphere from the atmosphere itself, and one from the water column. The correct answer is (a) 10 m.