A trough is 5 meters long, 1 meters wide, and 4 meters deep. The vertical cross-section of the trough parallel to an end is shaped like an isoceles triangle (with height 4 meters, and base, on top, of length 1 meters). The trough is full of water (density 1000 kg/m3 ). Find the amount of work in joules required to empty the trough by pumping the water out of an outlet that is located 3 meters above the top of the tank. See problem 21, page 464 of the text for a diagram of this trough configuration. (Note: Use g=9.8 m/s2 as the acceleration due to gravity.)
To find the amount of work required to empty the trough, we need to calculate the potential energy of the water that needs to be pumped out.
Let's begin by finding the volume of the trough. The volume of a trough can be calculated by multiplying the length, width, and depth.
Volume of trough = length * width * depth
Volume of trough = 5 m * 1 m * 4 m
Volume of trough = 20 cubic meters
Next, we need to find the mass of the water in the trough. The mass can be calculated by multiplying the volume of water by its density.
Mass = volume * density
Mass = 20 m^3 * 1000 kg/m^3
Mass = 20,000 kg
Now, let's calculate the height of the water column in the trough. The water column's height will be equal to the depth of the trough, which is 4 meters.
Height of water column = 4 meters
To calculate the potential energy of the water, we need to determine the vertical distance the water needs to be lifted. The water needs to be lifted from the top of the trough to the outlet, which is 3 meters above the top of the tank.
Vertical distance = height of water column + vertical distance to the outlet
Vertical distance = 4 meters + 3 meters
Vertical distance = 7 meters
Now we can calculate the potential energy using the formula:
Potential energy = mass * gravitational acceleration * vertical distance
Potential energy = 20,000 kg * 9.8 m/s^2 * 7 meters
Potential energy = 1,372,000 joules
Therefore, the amount of work required to empty the trough is 1,372,000 joules.