A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft³.) (ft) A) 62.4 lb B) 873.6 lb C) 436.8 lb D) 124.8 lb

Question

Quizplus · Accepted Answer

The force exerted by the water on the plate is equal to the weight of the water displaced by the plate. Since the plate is submerged vertically, it displaces a volume of water equal to the area (in square feet) of the plate times its depth (in feet) below the surface of the water. The depth of the plate is not given, but it does not matter since the force exerted by the water on the plate is proportional to the depth.

Let's assume that the plate has a height of 1 foot (this does not affect the result). Then the area of the plate is 1 ft times the length of the plate (which is not given, but also does not matter since the force exerted by the water on the plate is proportional to the area). Let's assume that the length of the plate is 1 foot (again, this does not affect the result). Then the area of the plate is 1 square foot.

Now, the weight of the water displaced by the plate is equal to the weight of the volume of water that is displaced. The volume of water displaced is equal to the area of the plate times the depth of the plate. Let's call the depth of the plate d.

So the weight of the water displaced by the plate is:

weight = volume x weight density

volume = area x depth = 1 ft^2 x d ft = d ft^3

weight = d ft^3 x 62.4 lb/ft^3 = 62.4d lb

Therefore, the force exerted by the water on the plate is 62.4d lb, which is proportional to the depth of the plate. Since the depth of the plate is not given, we cannot give a numerical answer. However, we can say that the answer choices that are proportional to the depth of the plate are C and D.

To determine the correct choice, we need to use the fact that the force exerted by the water on the plate is proportional to the depth of the plate. Since the plate is submerged vertically, the depth of the plate is the same as the height of the water above the plate (measured in feet). This is equal to the y-coordinate of the point where the plate intersects the y-axis.

Looking at the graph, we can see that the y-coordinate of this point is about 7 feet. Therefore, the force exerted by the water on the plate is proportional to 7 feet, which means that it is 7 times the weight of the water displaced by the plate. From the formula we derived earlier, we know that the weight of the water displaced by the plate is 62.4d lb, where d is the depth of the plate in feet. Since d = 7 feet, we have:

weight = 62.4 x 7 lb = 436.8 lb

Therefore, the force exerted by the water on the plate is 7 times the weight of the water displaced by the plate, which is 7 x 62.4 x 7 lb = 436.8 lb. Therefore, the correct choice is C.

A Vertical Plate Is Submerged in Water (The Surface of the Water