Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).
b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.

Verified step by step guidance

1. Understand that the work required to pump water out of the tank depends on the volume of water and the distance each water element must be lifted.
2. Set up the coordinate system with the vertex of the cone at the bottom (y=0) and the top of the tank at y=6 m. The radius of the water surface at height y is proportional to y, given by the similarity ratio \(r(y) = \frac{1.5}{6} y = 0.25 y\).
3. Express the volume of a thin horizontal slice of water at height y with thickness \(dy\) as \(dV = \pi r(y)^2 dy = \pi (0.25 y)^2 dy = \pi \times 0.0625 y^2 dy\).
4. The work to lift this slice to the top (y=6) is \(dW = \rho g dV (6 - y)\), where \(\rho\) is the density of water and \(g\) is acceleration due to gravity. Integrate \(dW\) from \(y=0\) to \(y=6\) to find the total work when the tank is full.
5. Repeat the integration from \(y=0\) to \(y=3\) (half the depth) to find the work when the tank is half full. Compare the two results to determine if the work for half the depth is half the work for the full tank.

Key Concepts

Work Done by Pumping Water

Work in this context is the energy required to move water against gravity from the tank to the top. It is calculated by integrating the force (weight of water) times the distance each water layer is lifted. Since the force depends on volume and height, work is not simply proportional to volume.

Volume of Water in a Conical Tank

The volume of water in a cone depends on the height of the water level. Because the radius changes linearly with height, the volume varies with the cube of the height. This nonlinear relationship affects how much water is present at half the depth compared to full.

Relationship Between Height and Radius in a Cone

In a cone, the radius at any height is proportional to that height due to similar triangles. For this tank, radius r = (1.5/6) * height. This relationship is essential to express volume and work integrals in terms of height, enabling calculation of work done for different water depths.