Final answer:
The work required to pump half of the water out of an aquarium that is 2 m long, 1 m wide, and 1 m deep is 4900 Joules. A Riemann sum can approximate this by dividing the water into thin slices and summing the work needed to lift each slice.
Step-by-step explanation:
To calculate the work needed to pump half of the water out of an aquarium that is 2 m long, 1 m wide, and 1 m deep, we consider the force needed to lift the water and the distance it must be moved. The volume of half the water in the aquarium is 1 cubic meter (since the full aquarium would contain 2 cubic meters of water), and the mass of this water is 1000 kg, using the density of water (1000 kg/m³).
The work done to lift this mass of water (W) can be calculated using the formula W = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height the water is lifted, which in this case is 0.5 meters (half the depth of the tank).
The work done is therefore W = 1000 kg * 9.8 m/s² * 0.5 m = 4900 Joules. To approximate the required work by a Riemann sum, we would divide the water into thin horizontal slices, calculate the work for each slice, and sum these amounts.
Each slice has a volume ΔV, and the work to move one slice is ΔW = ρgΔVh, where ρ is the density of water, g is the gravitational constant, ΔV is the volume of the slice, and h is the height to which the slice is lifted. Summing the work for all slices gives us a Riemann sum approximation of the total work.