For a marble track, it was necessary to find right triangles where the short leg and hypotenuse were integer lengths and where the longer leg�s length was an integer multiple of 6/5 or 1.2. The first two such triangles found had legs of length 4 and 15.6 and a hypotenuse of length 16, the second sides of length 3, 16.8, and 17. We will use a and b for the lengths of the shorter and longer legs, respectively, and c for the length of the hypotenuse. For a perfect right triangle, the Pythagorean theorem says a2 + b2 = c2 but this equation doesn�t hold for these triangles, for example, 32 + 16.82 = 291.24 not 172 = 289. To measure how far a triangle is from being a right triangle, it would be reasonable to see how far angle C, opposite the hypotenuse, is from 90 degrees. For this we use the Law of Cosines c2 = a2 + b2 � 2 a b cos C, so cos C = (a2 + b2 � c2)/(2 a b).
In our example, cos C = 0.0222� giving C = 88.73 degrees.
Another important factor is the slope of marble track, which is the ratio a/b. For a gradual slope we would like this to be as small as possible.
The table below lists some possible triangles, their sides, angle C, and their slopes. Only triangles with slopes less than ½ were chosen. To find the entries, a multiple of 1.2 was chosen for side b, then we looked at integer squares larger than b2. These gave us values for c. For a given b, c then a was chosen to be the closest integer to the square root of (c2 � b2). For example, if b = 20.4 (= 17 (1.2)) then b2 = 416.16. The next larger square is 441 which is 21 squared so c = 21. Finally 441-416.16 = 24.84 which is (very) close to 5 squared. If we take c=22, then c2 � b2 = 484 �416.16 = 67.84 which is close to 82. Taking c = 23 leads to a slope greater than ½.
Side a Side b Side c angle C Slope(a/b)
4 14.4=12(1.2) 15 90.82 .2778
7 14.4 16 89.90 .4861
4 15.6=13(1.2) 16 88.46 .2564
7 15.6 17 89.12 .4487
3 16.8 17 88.73 .1786
6 16.8 18 91.64 .3571
6 18 19 90.27 .3333
6 19.2 20 88.85 .3125
5 20.4 21 89.96 .2381
8 20.4 22 90.67 .3922
4 21.6 22 90.48 .1852
8 21.6 23 89.74 .3704
10 21.6 24 91.25 .4630
Side a Side b Side c Angle C Slope(a/b)
3 22.8 23 90.07 .1316
7 22.8 24 91.29 .3070
10 22.8 25 90.65 .4386
7 24 25 90.00 .2917
10 24 26 90.00 .4167
(the above two are actually right triangles)
6 25.2 26 90.94 .2381
10 25.2 27 89.32 .3922
12 25.2 28 90.47 .4762
6 26.4 27 89.28 .2273
9 26.4 28 90.73 .3409
12 26.4 29 90.004 .4545
5 27.6 28 89.43 .1812
9 27.6 29 89.80 .3261
12 27.6 30 89.50 .4348
3 28.8 29 90.85 .1042
11 28.8 31 90.95 .3819
14 28.8 32 89.90 .4861
8 30 31 89.64 .2667
11 30 32 90.26 .3667
14 30 33 89.52 .4667
7 31.2 32 90.20 .2244
11 31.2 33 89.55 .3526
John L. Drost
Professor of Mathematics
Marshall University
|