A trough is 8 meters long, 3 meters wide, and 5 meters deep. The vertical cross-section of the trough parallel to an end is shaped like an isoceles triangle (with height 5 meters, and base, on top, of length 3 meters). The trough is full of water (density ). Find the amount of work in joules required to empty the trough by pumping the water over the top. (Note: Use 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 in the trough and use the formula for work.
First, let's find the volume of water in the trough. The trough has the shape of an isosceles triangle when viewed from the end. The area of a triangle can be found using the formula:
Area = (base * height) / 2.
In this case, the base is 3 meters and the height is 5 meters, so the area of the triangular cross-section is:
Area = (3 * 5) / 2 = 7.5 square meters.
Since the trough is 8 meters long, the volume of water in the trough can be found by multiplying the area of the cross-section by the length of the trough:
Volume = Area * Length = 7.5 * 8 = 60 cubic meters.
Next, we'll find the mass of the water in the trough. The density of water is 1000 kg/m^3. So, the mass of the water is:
Mass = Density * Volume = 1000 * 60 = 60000 kg.
Now, we can calculate the potential energy of the water. The potential energy of an object is given by the formula:
Potential Energy = Mass * Acceleration due to Gravity * Height.
In this case, the height is 5 meters, and the acceleration due to gravity is denoted by "g". So, the potential energy of the water in the trough is:
Potential Energy = 60000 * g * 5.
Finally, we can calculate the amount of work required to empty the trough using the formula for work:
Work = Potential Energy.
Therefore, the amount of work in joules required to empty the trough by pumping the water over the top is 60000 * g * 5 joules.