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).
a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?

Verified step by step guidance

1. Identify the variables and set up the coordinate system: Let the vertical axis y measure the height from the tip (vertex) of the cone upwards, with y = 0 at the tip and y = 6 m at the top of the tank.
2. Express the radius of the water surface at height y: Since the tank is a cone, the radius r at height y varies linearly from 0 at y = 0 to 1.5 m at y = 6 m. Use similar triangles to write the radius as a function of y: \(r(y) = \frac{1.5}{6} y = 0.25 y\).
3. Write the volume of a thin horizontal slice of water at height y: The slice has thickness \(dy\) and radius \(r(y)\), so its volume is \(dV = \pi [r(y)]^2 dy = \pi (0.25 y)^2 dy = \pi \times 0.0625 y^2 dy\).
4. Determine the work required to pump this slice of water to the top: The weight density of water is \(\rho g\) (where \(\rho\) is the density of water and \(g\) is acceleration due to gravity). The distance the slice must be lifted is \$6 - y\(. So the work to move this slice is \)dW = \rho g \times dV \times (6 - y)$.
5. Set up the integral for total work: Integrate \(dW\) from \(y = 0\) to \(y = 6\) to find the total work required to pump all the water out of the tank: \(W = \int_0^6 \rho g \pi (0.0625 y^2) (6 - y) dy\). This integral can then be evaluated to find the total work.

Key Concepts

Work Done by a Variable Force

Work is the integral of force over distance. When pumping water from a tank, the force varies with the weight of the water slice being moved and the distance it must be lifted. Calculating work involves integrating the product of force and displacement over the height of the tank.

Volume of a Thin Slice of Water in a Cone

To find the work done, the tank is divided into thin horizontal slices of water. Each slice is a thin disk whose radius depends on its height in the cone. Using similar triangles, the radius at any height can be expressed as a function of height, allowing calculation of the slice's volume.

Relationship Between Radius and Height in a Cone

The radius of the water slice changes linearly with height due to the conical shape. Using the given dimensions, the radius at height y can be found by proportionality: radius = (1.5/6) * y. This relationship is essential to express the volume and weight of each slice as functions of height.