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I knew once I sat down with a pencil and paper
the answer would become "obvious".
it can be solved without calculus, using just basic
trigonometry.
the hard part is finding the area bounded by
the circle and the surface of the gas. multiplying
by the length of the cylinder yields volume.
1: draw a circle.
2: draw the surface of gas. a chord from one
point on the edge of the circle to another.
call the length of this chord "2 * Y"
this length is currently unknown.
3: draw lines from these 2 points to the center
of the circle. they go from the edge of the circle
to the center--radii, this length is known, call it "R"
4: call the angle between these 2 radii, "theta"
this is unknown
5: we know the height of gas from our dipstick.
we call this height "h".
6: the distance of the gas surface to the center
of the circle is given by:
X = R - h
7: make a poor attempt at an ascii drawing:
.---.
.-' '-.
.' '. . theta
/ \ /|\
| | R / | \ R
| o | / |X \
\ /|\ / / | \
' R /x| \ . /________|________\
'-. /--|-y\ .-' y |h y
'- h|_._' __|__
8: notice that area we are trying to find "Agas"
is the area of a pie shaped wedge minus the
area of a triangle:
Agas = Apie - Atri
9: simple trig:
Apie = 1/2 * R^2 * theta
Atri = X * R sin theta/2
10:
more trig:
theta = 2 * cos-1( X/R )
11:
we know X, we know R. we now know theta.
12: we now can calculate Apie, Atri, and Agas.
13: simplify the equations in terms of our known values:
Agas = R^2 * cos-1( R-h / h ) - (R - h) * sqrt( 2 R h - h^2 )
14: I had a beer at dinner, and it is after my bedtime.
check my math before deciding to head into the desert
without filling up.
--jeff
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