Final answer:
The depth of the lake where the pressure at half the depth is equal to 2/3 pressure at the bottom is approximately 30 meters. This calculation is based on the linear relationship between pressure and depth in a fluid and Pascal's Principle.
Step-by-step explanation:
The question is concerned with the relationship between pressure and depth in a fluid, which is a fundamental concept in physics, particularly fluid mechanics. According to Pascal's Principle, pressure in a fluid at rest is exerted equally in all directions and increases linearly with depth due to the weight of the fluid above. Moreover, this pressure is independent of the total volume of the fluid and depends only on the vertical distance below the surface. From the reference information, at a depth of 10.3 m in water, the total pressure is twice the atmospheric pressure, indicating that pressure increases by one atmosphere every 10.3 meters in depth. To find the depth of a lake, given that pressure at half the depth is 2/3 the pressure at the bottom, we can use this linear pressure-depth relationship.
If the pressure at the bottom of the lake corresponds to the depth D (unknown), then at half the depth (½D) the pressure is 2/3 of the pressure at the bottom. Since pressure at a depth of 10.3 meters equals twice the atmospheric pressure, and we know that pressure increases by one atmosphere for each additional 10.3 meters, we can set up a proportion to solve for the depth of the lake. As such, the pressure at the entire depth D would be equivalent to three atmospheres (since ½D equals two atmospheres), and thus D would equal 3 times 10.3 meters, which equals 30.9 meters. Given the context, we can approximate this to 30 meters to match the format of the question.