Final answer:
The pressure at a depth 2d below the surface of a lake, when at depth d the pressure is 2P atm, is 3P atm. This accounts for both the pressure due to the weight of the water and the atmospheric pressure.
Step-by-step explanation:
The student's question asks: If the pressure at a depth d below the surface of a lake is 2P atm (twice atmospheric pressure), the pressure at a depth 2d is To answer this, we must understand how pressure works within a fluid such as a lake. According to Pascal's Principle, the pressure at a given depth is equal to the weight of the water above it plus the atmospheric pressure on the water's surface. For example, at a depth where the pressure due to the water's weight is 1 atm, the total pressure (including the atmospheric pressure) would be 2 atm. Therefore, if at depth d the total pressure is 2P atm, then at depth 2d the pressure due to the water's weight would be twice that of depth d, meaning another 2P atm due to the water, added to the atmospheric pressure P atm. This results in a pressure of 3P atm at depth 2d. So, the correct answer is 3) 3P atm.