Question

A spherical tank is 3 m tall and therefore has a capacity of about 14000 l (note that 3m is its diameter, and one cubic meter is 1000 l). You have been asked to put level lines on the tank showing the depths where the tank contains 1000, 5000, and 10000 l of liquid.

Answer 1

1) The level line for 1000 liters of liquid is set at a height of 0.486 meters.

2) The level line for 5000 liters of liquid is set at a height of 1.203 meters.

3) The level line for 10000 liters of liquid is set at a height of 1.926 meters.

How to determine label heights for a spherical tank

Firstly we derive an expression for the volume of a spherical section ($V$), in cubic meters, in terms of tank radius ($R$), in meters, and liquid height ($h$), in meters. The volume can be found by using the slicing method integral formula:

$V = \int\limits^{H}_{0} {A(h)} \, dh$ (1)

Where:

• $A(h)$ - Area of the infinitesimal circle, in square meters.
• $H$ - Required height, in meters.

If we know that $A = \pi \cdot [R^{2}-(R-h)^{2}]$, then the volume formula is:

$V = \pi\int\limits^{H}_{0} {h\cdot (2\cdot R - h)} \, dh$

$V = 2\pi\cdot R\int\limits^{H}_{0} {h} \, dh - \pi \int\limits^{H}_{0} {h^{2}} \, dh$

$V = \pi\cdot R\cdot H^{2}-\frac{\pi\cdot H^{3}}{3}$

Then the following third order polynomial is found:

$\frac{\pi\cdot H^{3}}{3} -\pi\cdot R\cdot H^{2}+V = 0$ (2)

Now we proceed to solve numerically for the following three cases:

Case 1 - $R = 1.5\,m$, $V = 1\,m^{3}$

$H = 0.486\,m$

The level line for 1000 liters of liquid is set at a height of 0.486 meters. $\blacksquare$

Case 2 - $R = 1.5\,m$, $V = 5\,m^{3}$

$H = 1.203\,m$

The level line for 5000 liters of liquid is set at a height of 1.203 meters. $\blacksquare$

Case 3 - $R = 1.5\,m$, $V = 10\,m^{3}$

$H = 1.926\,m$

The level line for 10000 liters of liquid is set at a height of 1.926 meters. $\blacksquare$

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