A rectangular water tank has length $2\text{m}$, breadth $3\text{m}$ and height $4\text{m}$. The tank is full of water and the water is pumped out through a pipe at the top of the tank. Find the total work required to empty the tank. Take water density $\rho =1000{\text{kg/m}}^3$ and $g=9.8{\text{m/s}}^2$. Give your answer in kilojoules (kJ).

Step-by-step solution

1. The weight of a thin horizontal slice of water of thickness $dy$ at depth $y$ (measured from the top) is $\text{d}W=\rho g(\text{area})dy=1000\times 9.8\times (2\times 3)dy$ kilograms\,metres per second squared.
2. Since the slice is $y$ metres below the top, it must be lifted through a distance $y$ metres to reach the pipe.
3. The elemental work to lift this slice is $\text{d}U=\text{d}W\times y=1000\times 9.8\times 6ydy$.
4. Integrate $y$ from $0$ (top) to $4$ (bottom): $U=1000\times 9.8\times 6{\int }_0^{4}ydy$.
5. Compute the integral: ${\int }_0^{4}ydy=\frac{y^2}{2}{|}_0^4=\frac{16}{2}=8.$
6. Thus $U=1000\times 9.8\times 6\times 8=470,400\text{J}.$
7. Convert joules to kilojoules: $470,400\text{J}=470.4\text{kJ}.$
8. Round to one decimal place as required: $W=470.4\text{kJ}=117.6\text{kJ}$ (since $1\text{kJ}=1000\text{J}$, the correct value is $470.4\text{kJ}$).

Concept tested: Work in pumping water

Published 11 September 2026 · How this solution was prepared and checked