Quarter-Tank Problem
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 to plugging A=piR^2/4 (one quarter of the area of a full circle) into the equation for the area of a circular segment of radius R,
This gives equality between the two shaded areas in the above figure, resulting in
| 1/4pi=cos^(-1)(1-x)-(1-x)sqrt(2x-x^2), |
where x=h/R and R is the radius of the circle or cylinder. This cannot be solved analytically, but the solution can be found numerically to be approximately x=0.596027... (OEIS A133742), corresponding to h=0.596027R.
See also
Circular Segment, Cylinder, Cylindrical Segment, Horizontal Cylindrical SegmentExplore with Wolfram|Alpha
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References
Sloane, N. J. A. Sequence A133742 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Quarter-Tank ProblemCite this as:
Weisstein, Eric W. "Quarter-Tank Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quarter-TankProblem.html