Math CentralQuandaries & Queries


Question from Philip, a parent:

I have a steel gas tank that is 3' dia X 5' length. The total volume is 1000 litres.
But how much is left when I use a stick and measure 6" from the bottom or 12" or 24" ??
Is there a formula to use for this task?


Hi Philip,

Nowhere but here in Canada would we mix feet, inches and litres in the same problem.

We have answered similar questions before and probably the best expression in this response to Jason. The expression is

\[V = L \left(r^2 cos^{-1}\frac{r - h}{r} - (r - h) \sqrt{2 h r - h^2} \right) \]

where $V$ is the volume, $L$ is the length, $r$ is the radius and $h$ is the height of the liquid in the tank. I'm going to use inches so for your tank $L = 60$, $r = 18$ and the volume $V$ will be in cubic inches. Thus the volume when the height of the liquid is $h$ inches is

\[V = 60 \left(324 \; cos^{-1}\frac{18 - h}{18} - (18 - h) \sqrt{36 h - h^2} \right) \mbox{cubic feet.}\]

The one caution in applying this expression is that you make sure your calculator is in radian mode to find $\cos^{-1},$ not degree mode.

Thus, for example if $h = 6$ inches then

\[V = 60 \left(324 \; cos^{-1}\frac{18 - 6}{18} - (18 - 6) \sqrt{36 \times 6 - 6^2} \right) \mbox{cubic feet.}\]

I got $V = 6690.56$ cubic inches.

To convert the volume to liters you can ask Google. If you type 6690.56 cubic inches in liters into the Google search window the response is 6 690.56 (cubic inches) = 109.638635 liters.


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