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ProfessionalEngineering# Level of liquid in the cylindrical tank, AKA Quarter-Tank Problem

##### This online calculator finds the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be full at specified percentage (for example, one quarter full amounts)

The Quarter-Tank Problem can be formulated like this: Finding the height above the bottom of a horizontal cylinder (such as a cylindrical gas tank) to which the it must be filled for it to be one quarter full amounts.

This calculator solves the more general problem, it finds the height above the bottom of a horizontal cylinder to which the it must be filled to be full at specified percentage.

This task is the exact opposite to the task solved in Cylindrical tank volume calculator.

Here is the equation which connects level of liquid and volume of liquid in the tank:

\pi R^2 p = R^2 acos((R - h) / R) - (R - h)\sqrt{(2Rh - h^2)}

where p is needed fraction, i.e. 0.25 for one quarter full, R is radius of cylinder and h is height of liquid.

This equation does not have analytical solution for h, however, it can be solved by numerical methods, like Secant method, which is indeed used in this calculator.

Also it is worth to note that this is quite simple case, however, we also have calculators for tilted cylinder case - Liquid level in the tilted cylinder and Tilted cylindrical tank volume

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