The work required to pump the water out of the spout.
The work is calculated by using the formula W=∫abf(x)dx
Let the circular disk slice of water have thickness ∆x m .
It is lying at the coordinate x with radius r=9-x2 m
The given tank is half full of oil, therefore, the lower limit changes from -3 to 0 and the density of oil is 900 kg/m3
Volume of the slice of oil =πr2 ∆x m3=π9-x22∆x m3=π9-x2∆x m3
Therefore, the weight of the slice is =g·density·volume=9.8×900π9-x2∆x N
This weight is lifted by the pump about x+4m.
Therefore, the work needed to pump the water out of the spout is
Therefore, the total work reqiured is
The work required to pump the water out of the spout is ≈2.55*106 J