Filling a Cylindrical Tank

Relating Inflow to Water-Level Rise

A tank can receive water at a rate measured in units of volume per time, such as cubic metres per minute. The quantity of practical interest may instead be how quickly the water surface rises, measured in metres per minute. For a cylindrical tank with a fixed radius, a simple volume formula connects these two rates.

The key feature of a cylinder is that each horizontal slice has the same area. Consequently, adding a given volume raises the water by the same amount whether the tank is nearly empty or already partly full, provided the tank has vertical sides and the radius remains fixed.

The Volume of Water in a Cylindrical Tank

Let \(r\) be the tank’s fixed radius and let \(h(t)\) be the water depth at time \(t\). The water occupies a cylinder of radius \(r\) and height \(h(t)\). Since the volume of a cylinder is its base area multiplied by its height, the water volume \(V(t)\) is

The inflow rate is the rate at which the water volume increases, \(dV/dt\). The water-level rate is \(dh/dt\). These are different quantities: one measures volume change per time, while the other measures height change per time.

This model assumes that the tank’s sides are vertical, the radius does not change, and the water surface is horizontal. It applies while the water remains inside the cylindrical part of the tank. If water spills out, the volume need not continue increasing at the incoming flow rate.

The Related-Rates Equation

Because the radius is fixed, \(\pi r^2\) is constant. Differentiate the volume equation with respect to time. The chain rule accounts for the fact that the depth depends on time:

If the inflow rate \(Q\) is known at a particular instant, then \(dV/dt=Q\) at that instant. Solving the equation for the rate of rise gives the main formula.

Why the Formula Works

The cross-sectional area of the tank is \(A=\pi r^2\). A small rise in water level by an amount \(\Delta h\) adds a thin layer whose volume is \(A\Delta h\). Thus, for a cylindrical tank, a change in depth and the corresponding change in volume satisfy \(\Delta V=A\Delta h\). Dividing by a time interval \(\Delta t\) gives

Taking the limit as the time interval shrinks to zero gives \(dV/dt=A\,dh/dt\), which is the differentiated volume equation. Because the area \(A\) is constant throughout the tank, the same amount of incoming volume produces the same level-rise rate at every depth, as long as the inflow rate itself is unchanged.

Worked Examples

Common Mistake: Using the Diameter as the Radius

A frequent error is to substitute the diameter directly into the area formula. For example, if the diameter is \(10\) centimetres, an incorrect calculation is

This treats \(10\) centimetres as the radius. But the radius is half the diameter, so the correct area is

With an inflow of \(250\text{ cm}^3/\text{s}\), using the incorrect area would give \(2.5/\pi\text{ cm}/\text{s}\), whereas the correct rate is \(10/\pi\text{ cm}/\text{s}\). The incorrect value is four times too small because doubling the radius multiplies the circular area by four.

Applying the Model Carefully

The rate formula separates the roles of the tank and the flow: the inflow \(Q\) describes how quickly volume is added, while the area \(\pi r^2\) converts that volume rate into a depth rate. A larger radius means a larger cross-sectional area, so the same inflow raises the water more slowly.

Before calculating, identify the radius rather than the diameter, express all lengths and volume rates in compatible units, and decide whether the given rate is the net change in volume. If water enters and also drains away, the volume rate is the inflow minus the outflow. Finally, check that the answer has units of length per time and that multiplying it by the cross-sectional area recovers the volume rate.

For a cylindrical tank of fixed radius, the water-level rate does not depend on the current depth. That conclusion depends on the cylinder’s constant cross-section; it should not be assumed for a tank whose width changes with height.