TURPENTINE, 23 
in which 
«= factor 3.1416. 
r=radius of tank (4 average inside diameter). 
M= angle subtended by the chord whose length is C, which is 
the base of the segment A. 
h= height of the segment, or the measured outage from the top 
of the car. 
Taking as a definite example for solution a tank car having the 
following dimensions: Length of body, 32 feet 4 inches; regular 
extensions inside, 1 inch each; average inside diameter, 78 inches; 
diameter of dome, 44 inches; and an outage of 6 inches, 
r =4 inside diameter = 39 inches. 
h =outage = 6 inches. 
L,=total cylindrical length of tank = 388 inches + 2 inches 
_* = 390 inches. 
The first step is to calculate the angle VW. If the line ad is drawn 
bisecting the angle WM, it will be perpendicular to the chord bc, and 
the triangles aec and aeb will be right angled at the point e. In the 
triangle aec two sides are known, namely, side ac = radius = 39 
inches, and side ae, the difference between the whole line ad (radius) 
and the height of the segment ed (h = 6 inches), making side ae = 33 
inches. These two lines are, respectively, the hypotenuse and the 
adjacent side of the angle cae (4 of the whole angle MV). In any 
right-angle triangle, the cosine of any acute angle is equal to the 
quotient of the side adjacent to the angle divided by the hypotenuse. 
In this case, the cosine of angle cae = 0.8461. <A table of 
_ natural trigonometric functions shows that 0.8461 is the cosine of 
angle 32° 13’, or, expressed decimally, 32.22°. If angle cae is 32.22°, 
the whole angle cab, or M, is 64.44°. 
- The next step is to calculate the length of the chord (@, or line be, 
in figure 3. This can be done in two ways, the simpler of which is 
here given. In any right-angle triangle, such as triangle aec, the 
square of the hypotenuse (line ac) is equal to the sum of the square of 
the other two sides. Then 
ac= ae+ec 
or transposing, 
eC = aC? — ae? 
ac = 39 inches, or ac?= 1,521 square inches 
ae = 33 inches, or ae?= 1,089 square inches 
then ec?= 1,521 — 1,089 = 432 square inches, 
and, extracting square root, ec = 20.8 inches (approximately). 
Since ¢c is only half of the entire chord, the whole chord @ (line bc) 
C 
= 41.6 inches, and zx = 20.8 inches. 
