Final answer:
The work required to pump all the water over the top of a spherical tank that is halfway filled, with a radius of 2, is 8π, calculated through the integration of work for each thin slice of water. Therefore the correct answer is c. 8π.
Step-by-step explanation:
The student is asking how much work is required to pump all the water over the top of a spherical tank that is halfway filled, where the tank has a radius of 2. To solve this, we assume the density of the water is constant and use the physical concept of work (W = force x distance).
The work done to lift a thin slice of water to the top of the tank can be expressed as dW = ρghdV, where ρ is the density of water, g is the acceleration due to gravity, h is the height above the water slice to the top of the tank, and dV is the volume of the water slice.
The total work is the integral of dW from the bottom of the filled part of the tank to the top. The correct answer for the total work needed, using calculus and symmetry considerations, is 8π (in terms of the unit of force times unit of distance, such as joules if using SI units).