A hollow cubical box is 0.30 m on an edge. This box is floating in a lake with one-third of its height beneath the surface. The walls of the box have a negligible thickness. Water from a hose is poured into the open top of the box. What is the depth of the water in the box just at the instant that water from the lake begins to pour into the box from the lake?
Question
A hollow cubical box is 0.30 m on an edge. This box is floating in a lake with one-third of its height beneath the surface. The walls of the box have a negligible thickness. Water from a hose is poured into the open top of the box. What is the depth of the water in the box just at the instant that water from the lake begins to pour into the box from the lake?
Solution
To solve this problem, we need to understand that the box will start to sink when the weight of the water inside the box is equal to the weight of the water displaced by the box.
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First, we need to calculate the volume of the water displaced by the box when it's floating with one-third of its height beneath the surface. The volume of the submerged part of the box is 0.30m * 0.30m * (0.30m/3) = 0.03 cubic meters.
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The weight of the water displaced by the box is equal to the volume of the water displaced multiplied by the density of water (1000 kg/m^3) and the acceleration due to gravity (9.8 m/s^2). So, the weight of the water displaced is 0.03 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 294 N.
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Now, we need to calculate the depth of the water in the box when the weight of the water in the box is equal to 294 N. The weight of the water in the box is equal to the volume of the water in the box multiplied by the density of water and the acceleration due to gravity.
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Setting this equal to 294 N and solving for the volume gives us a volume of 0.03 m^3.
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Since the box is a cube, the depth of the water in the box is the cube root of the volume, which is approximately 0.31 m.
So, the depth of the water in the box just at the instant that water from the lake begins to pour into the box from the lake is approximately 0.31 m.
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