ΔAB
nC is isosceles with vertex A.
B1, B2, ..., Bn-1 are unique points alternating between the legs of the triangle where
AB1 = B1B2 = B2B3 = ... = Bn-1Bn = BnC.
If ∠CABn = 0.8°, find n.
Let the length of AB1 be the unit of length. The base is of length 1. Since the vertex at A has an angular measure of 0.8°, the height of the skinny triangle is .5/tan(0.4°) ~= 71.6185608347376, taking one side (one half) of the isosceles triangle as a right triangle.
Segment AB1 is the only one that lies along one side of the triangle. After that the 1-unit segments span from side to side of the triangle, forming small isosceles triangles along the way, with their bases lying along one side or the other of the larger triangle, alternating sides.
The first such smaller triangle, the only one that has a side along one of the sides of the large triangle, has base angles of 0.8°, as one of them is coincident with the apex of the large triangle. Thus its vertex angle is 180 - 2*.8 = 178.4°. Using the law of cosines, its base is sqrt(1+1-2*cos(178.4°)) ~= 1.99980504801861. That is the first distance measured down along one side of the large triangle, from A to B2, with B1 having been at the apex of this first triangle. The height attained thus far being cos(0.4°) times the slant height, with the slant height being the accumulated bases of the small isosceles triangles on just one side or the other in alternate values of k along the way to n.
From then on, each small isosceles triangle faces the opposite way. The first time only, the new base angle is the supplement of the preceding apex angle; each time after that the new base angle is 180 minus the previous apex angle and minus the second previous base angle. For example, in the triangle that leads to B4 in the table below, the base angle of 2.4° is 180 - 176.8 - 0.8. The new apex angle is of course 180 minus twice the base angle.
The table below eventually reaches B112 at the full height of the large triangle, so n = 112. Incidentally since the even B's are on the opposite side from B1, that first segment lay on AC.
It's interesting that a rational number of degrees would lead to such a situation. Offhand I'd say it indicates it is more related to the circumference of a circle than to the radius or diameter. It creates a suspicion that it might not be exact and the zeros after the integral portion are misleading, but I've added more about that further down; the result is verified.
k base apex base slant height relative
angle angle height so far height
2 0.800 178.400 1.99981 1.9998050 1.9997563 0.0279223
3 1.600 176.800 1.99922 2.9992202 2.9991471 0.0418767
4 2.400 175.200 1.99825 3.9980507 3.9979533 0.0558229
5 3.200 173.600 1.99688 4.9961018 4.9959800 0.0697582
6 4.000 172.000 1.99513 5.9931788 5.9930328 0.0836799
7 4.800 170.400 1.99299 6.9890875 6.9889172 0.0975853
8 5.600 168.800 1.99045 7.9836336 7.9834391 0.1114716
9 6.400 167.200 1.98754 8.9766233 8.9764046 0.1253363
10 7.200 165.600 1.98423 9.9678630 9.9676201 0.1391765
11 8.000 164.000 1.98054 10.9571595 10.9568924 0.1529896
12 8.800 162.400 1.97646 11.9443198 11.9440287 0.1667728
13 9.600 160.800 1.97199 12.9291515 12.9288365 0.1805235
14 10.400 159.200 1.96714 13.9114627 13.9111237 0.1942391
15 11.200 157.600 1.96191 14.8910618 14.8906990 0.2079168
16 12.000 156.000 1.95630 15.8677579 15.8673712 0.2215539
17 12.800 154.400 1.95030 16.8413605 16.8409501 0.2351478
18 13.600 152.800 1.94392 17.8116799 17.8112459 0.2486959
19 14.400 151.200 1.93717 18.7785269 18.7780692 0.2621956
20 15.200 149.600 1.93003 19.7417129 19.7412318 0.2756441
21 16.000 148.000 1.92252 20.7010503 20.7005458 0.2890388
22 16.800 146.400 1.91464 21.6563519 21.6558242 0.3023773
23 17.600 144.800 1.90638 22.6074316 22.6068807 0.3156567
24 18.400 143.200 1.89775 23.5541039 23.5535299 0.3288747
25 19.200 141.600 1.88875 24.4961843 24.4955874 0.3420285
26 20.000 140.000 1.87939 25.4334892 25.4328694 0.3551156
27 20.800 138.400 1.86965 26.3658357 26.3651932 0.3681335
28 21.600 136.800 1.85955 27.2930421 27.2923770 0.3810797
29 22.400 135.200 1.84909 28.2149278 28.2142402 0.3939515
30 23.200 133.600 1.83827 29.1313128 29.1306029 0.4067466
31 24.000 132.000 1.82709 30.0420187 30.0412866 0.4194623
32 24.800 130.400 1.81555 30.9468678 30.9461136 0.4320963
33 25.600 128.800 1.80367 31.8456837 31.8449077 0.4446460
34 26.400 127.200 1.79142 32.7382913 32.7374935 0.4571091
35 27.200 125.600 1.77883 33.6245165 33.6236971 0.4694830
36 28.000 124.000 1.76590 34.5041865 34.5033456 0.4817654
37 28.800 122.400 1.75261 35.3771298 35.3762677 0.4939539
38 29.600 120.800 1.73899 36.2431763 36.2422931 0.5060461
39 30.400 119.200 1.72503 37.1021572 37.1012530 0.5180396
40 31.200 117.600 1.71073 37.9539049 37.9529800 0.5299322
41 32.000 116.000 1.69610 38.7982534 38.7973079 0.5417214
42 32.800 114.400 1.68113 39.6350381 39.6340722 0.5534050
43 33.600 112.800 1.66584 40.4640958 40.4631098 0.5649808
44 34.400 111.200 1.65023 41.2852651 41.2842590 0.5764464
45 35.200 109.600 1.63429 42.0983856 42.0973597 0.5877996
46 36.000 108.000 1.61803 42.9032991 42.9022535 0.5990382
47 36.800 106.400 1.60146 43.6998484 43.6987834 0.6101600
48 37.600 104.800 1.58458 44.4878783 44.4867942 0.6211629
49 38.400 103.200 1.56739 45.2672353 45.2661322 0.6320447
50 39.200 101.600 1.54989 46.0377673 46.0366454 0.6428033
51 40.000 100.000 1.53209 46.7993242 46.7981837 0.6534365
52 40.800 98.400 1.51399 47.5517574 47.5505986 0.6639424
53 41.600 96.800 1.49560 48.2949204 48.2937435 0.6743188
54 42.400 95.200 1.47691 49.0286681 49.0274733 0.6845638
55 43.200 93.600 1.45794 49.7528576 49.7516452 0.6946753
56 44.000 92.000 1.43868 50.4673477 50.4661179 0.7046514
57 44.800 90.400 1.41914 51.1719991 51.1707521 0.7144901
58 45.600 88.800 1.39933 51.8666744 51.8654104 0.7241895
59 46.400 87.200 1.37924 52.5512382 52.5499575 0.7337477
60 47.200 85.600 1.35888 53.2255570 53.2242599 0.7431629
61 48.000 84.000 1.33826 53.8894994 53.8881861 0.7524332
62 48.800 82.400 1.31738 54.5429359 54.5416067 0.7615569
63 49.600 80.800 1.29624 55.1857392 55.1843944 0.7705320
64 50.400 79.200 1.27485 55.8177839 55.8164237 0.7793570
65 51.200 77.600 1.25321 56.4389468 56.4375714 0.7880300
66 52.000 76.000 1.23132 57.0491068 57.0477166 0.7965493
67 52.800 74.400 1.20920 57.6481450 57.6467402 0.8049134
68 53.600 72.800 1.18684 58.2359446 58.2345255 0.8131206
69 54.400 71.200 1.16425 58.8123910 58.8109578 0.8211692
70 55.200 69.600 1.14143 59.3773718 59.3759248 0.8290578
71 56.000 68.000 1.11839 59.9307768 59.9293163 0.8367847
72 56.800 66.400 1.09513 60.4724982 60.4710245 0.8443485
73 57.600 64.800 1.07165 61.0024304 61.0009438 0.8517477
74 58.400 63.200 1.04797 61.5204700 61.5189708 0.8589808
75 59.200 61.600 1.02409 62.0265161 62.0250046 0.8660465
76 60.000 60.000 1.00000 62.5204700 62.5189464 0.8729434
77 60.800 58.400 0.97572 63.0022354 63.0007001 0.8796700
78 61.600 56.800 0.95125 63.4717184 63.4701717 0.8862252
79 62.400 55.200 0.92659 63.9288275 63.9272696 0.8926076
80 63.200 53.600 0.90176 64.3734735 64.3719048 0.8988159
81 64.000 52.000 0.87674 64.8055698 64.8039905 0.9048491
82 64.800 50.400 0.85156 65.2250321 65.2234426 0.9107059
83 65.600 48.800 0.82621 65.6317787 65.6301793 0.9163851
84 66.400 47.200 0.80070 66.0257302 66.0241212 0.9218856
85 67.200 45.600 0.77503 66.4068098 66.4051915 0.9272064
86 68.000 44.000 0.74921 66.7749434 66.7733161 0.9323465
87 68.800 42.400 0.72325 67.1300590 67.1284231 0.9373048
88 69.600 40.800 0.69714 67.4720874 67.4704432 0.9420804
89 70.400 39.200 0.67090 67.8009621 67.7993098 0.9466723
90 71.200 37.600 0.64453 68.1166188 68.1149589 0.9510797
91 72.000 36.000 0.61803 68.4189961 68.4173288 0.9553016
92 72.800 34.400 0.59142 68.7080349 68.7063606 0.9593374
93 73.600 32.800 0.56468 68.9836790 68.9819979 0.9631860
94 74.400 31.200 0.53784 69.2458746 69.2441871 0.9668469
95 75.200 29.600 0.51089 69.4945705 69.4928770 0.9703194
96 76.000 28.000 0.48384 69.7297184 69.7280191 0.9736026
97 76.800 26.400 0.45670 69.9512723 69.9495676 0.9766961
98 77.600 24.800 0.42947 70.1591890 70.1574793 0.9795991
99 78.400 23.200 0.40216 70.3534281 70.3517136 0.9823112
100 79.200 21.600 0.37476 70.5339517 70.5322328 0.9848318
101 80.000 20.000 0.34730 70.7007245 70.6990015 0.9871603
102 80.800 18.400 0.31976 70.8537140 70.8519874 0.9892964
103 81.600 16.800 0.29217 70.9928905 70.9911605 0.9912397
104 82.400 15.200 0.26451 71.1182268 71.1164937 0.9929897
105 83.200 13.600 0.23681 71.2296985 71.2279626 0.9945461
106 84.000 12.000 0.20906 71.3272837 71.3255455 0.9959087
107 84.800 10.400 0.18127 71.4109636 71.4092234 0.9970771
108 85.600 8.800 0.15344 71.4807218 71.4789799 0.9980511
109 86.400 7.200 0.12558 71.5365447 71.5348014 0.9988305
110 87.200 5.600 0.09770 71.5784213 71.5766770 0.9994152
111 88.000 4.000 0.06980 71.6063437 71.6045987 0.9998050
112 88.800 2.400 0.04188 71.6203062 71.6185608 1.0000000
clc
CAB=.8;
halfCAB=CAB/2;
halfBase=.5;
height=halfBase/tand(halfCAB);
baseAngle(1)=CAB;
slant(1)=1; slant(2)=0;
currHeight=cosd(halfCAB);
side=2;
n=1;
while currHeight<=height
vAngle=180-2*baseAngle(n);
base=sqrt(2-2*cosd(vAngle));
slant(side)=slant(side)+base;
currHeight=slant(side)*cosd(halfCAB);
n=n+1;
fprintf('%3d %6.3f %7.3f %7.5f %10.7f %10.7f %9.7f
' ,...
n,baseAngle(n-1),vAngle,base,slant(side),currHeight,currHeight/height)
side=3-side;
if n==2
baseAngle(n)=180-vAngle;
else
baseAngle(n)=180-baseAngle(n-2)-vAngle;
end
end
Being that I was suspicious of the setup as involving a rational number of degrees, I wanted to be sure the result was not a fluke of rounding error giving a ratio of exactly 1. (In fact the ratio of large triangle height to the height at which the trellis ended was actually 1.00000000000002, making me feel justified that the 2 at the end was just a rounding error.)
So I redid it with vpa, variable precision arithmetic:
clc
digits(50)
CAB=vpa(.8);
halfCAB=CAB/2;
halfBase=vpa(.5);
height=halfBase/vpa(tand(halfCAB));
baseAngle(1)=CAB;
slant(1)=vpa(1); slant(2)=vpa(0);
currHeight=vpa(cosd(halfCAB));
side=vpa(2);
n=1;
while currHeight<=height
vAngle=180-2*baseAngle(n);
base=sqrt(2-2*vpa(cosd(vAngle)));
slant(side)=slant(side)+base;
currHeight=slant(side)*vpa(cosd(halfCAB));
n=n+1;
fprintf('%3d %6.3f %7.3f %7.5f %10.7f %10.7f %9.7f
' ,...
n,baseAngle(n-1),vAngle,base,slant(side),currHeight,currHeight/height)
if currHeight/height>.9999
disp([currHeight currHeight/height]) % check on full accuracy
end
side=3-side;
if n==2
baseAngle(n)=180-vAngle;
else
baseAngle(n)=180-baseAngle(n-2)-vAngle;
end
end
disp([currHeight currHeight/height])
The results were exact to 50 places, the first 32 of which (the actual height) agreed with the 32 that are afforded by default when using vpa. The 112th iteration does exactly match the height of the skinny triangle.
An artifact of doing it over is that the ratio did not exceed 1 immediately, as B112 to B113 was in fact a horizontal line and n could be either B112 or B113 depending on which one you want to call C instead. After that, in reality the points would go back up the thin triangle, but the trigonometry assumed positive length bases of the small triangles and kept increasong the height, stopping the iterations.
k base apex base slant height relative
angle angle height so far height
2 0.800 178.400 1.99981 1.9998050 1.9997563 0.0279223
3 1.600 176.800 1.99922 2.9992202 2.9991471 0.0418767
4 2.400 175.200 1.99825 3.9980507 3.9979533 0.0558229
5 3.200 173.600 1.99688 4.9961018 4.9959800 0.0697582
6 4.000 172.000 1.99513 5.9931788 5.9930328 0.0836799
7 4.800 170.400 1.99299 6.9890875 6.9889172 0.0975853
8 5.600 168.800 1.99045 7.9836336 7.9834391 0.1114716
9 6.400 167.200 1.98754 8.9766233 8.9764046 0.1253363
10 7.200 165.600 1.98423 9.9678630 9.9676201 0.1391765
11 8.000 164.000 1.98054 10.9571595 10.9568924 0.1529896
12 8.800 162.400 1.97646 11.9443198 11.9440287 0.1667728
13 9.600 160.800 1.97199 12.9291515 12.9288365 0.1805235
14 10.400 159.200 1.96714 13.9114627 13.9111237 0.1942391
15 11.200 157.600 1.96191 14.8910618 14.8906990 0.2079168
16 12.000 156.000 1.95630 15.8677579 15.8673712 0.2215539
17 12.800 154.400 1.95030 16.8413605 16.8409501 0.2351478
18 13.600 152.800 1.94392 17.8116799 17.8112459 0.2486959
19 14.400 151.200 1.93717 18.7785269 18.7780692 0.2621956
20 15.200 149.600 1.93003 19.7417129 19.7412318 0.2756441
21 16.000 148.000 1.92252 20.7010503 20.7005458 0.2890388
22 16.800 146.400 1.91464 21.6563519 21.6558242 0.3023773
23 17.600 144.800 1.90638 22.6074316 22.6068807 0.3156567
24 18.400 143.200 1.89775 23.5541039 23.5535299 0.3288747
25 19.200 141.600 1.88875 24.4961843 24.4955874 0.3420285
26 20.000 140.000 1.87939 25.4334892 25.4328694 0.3551156
27 20.800 138.400 1.86965 26.3658357 26.3651932 0.3681335
28 21.600 136.800 1.85955 27.2930421 27.2923770 0.3810797
29 22.400 135.200 1.84909 28.2149278 28.2142402 0.3939515
30 23.200 133.600 1.83827 29.1313128 29.1306029 0.4067466
31 24.000 132.000 1.82709 30.0420187 30.0412866 0.4194623
32 24.800 130.400 1.81555 30.9468678 30.9461136 0.4320963
33 25.600 128.800 1.80367 31.8456837 31.8449077 0.4446460
34 26.400 127.200 1.79142 32.7382913 32.7374935 0.4571091
35 27.200 125.600 1.77883 33.6245165 33.6236971 0.4694830
36 28.000 124.000 1.76590 34.5041865 34.5033456 0.4817654
37 28.800 122.400 1.75261 35.3771298 35.3762677 0.4939539
38 29.600 120.800 1.73899 36.2431763 36.2422931 0.5060461
39 30.400 119.200 1.72503 37.1021572 37.1012530 0.5180396
40 31.200 117.600 1.71073 37.9539049 37.9529800 0.5299322
41 32.000 116.000 1.69610 38.7982534 38.7973079 0.5417214
42 32.800 114.400 1.68113 39.6350381 39.6340722 0.5534050
43 33.600 112.800 1.66584 40.4640958 40.4631098 0.5649808
44 34.400 111.200 1.65023 41.2852651 41.2842590 0.5764464
45 35.200 109.600 1.63429 42.0983856 42.0973597 0.5877996
46 36.000 108.000 1.61803 42.9032991 42.9022535 0.5990382
47 36.800 106.400 1.60146 43.6998484 43.6987834 0.6101600
48 37.600 104.800 1.58458 44.4878783 44.4867942 0.6211629
49 38.400 103.200 1.56739 45.2672353 45.2661322 0.6320447
50 39.200 101.600 1.54989 46.0377673 46.0366454 0.6428033
51 40.000 100.000 1.53209 46.7993242 46.7981837 0.6534365
52 40.800 98.400 1.51399 47.5517574 47.5505986 0.6639424
53 41.600 96.800 1.49560 48.2949204 48.2937435 0.6743188
54 42.400 95.200 1.47691 49.0286681 49.0274733 0.6845638
55 43.200 93.600 1.45794 49.7528576 49.7516452 0.6946753
56 44.000 92.000 1.43868 50.4673477 50.4661179 0.7046514
57 44.800 90.400 1.41914 51.1719991 51.1707521 0.7144901
58 45.600 88.800 1.39933 51.8666744 51.8654104 0.7241895
59 46.400 87.200 1.37924 52.5512382 52.5499575 0.7337477
60 47.200 85.600 1.35888 53.2255570 53.2242599 0.7431629
61 48.000 84.000 1.33826 53.8894994 53.8881861 0.7524332
62 48.800 82.400 1.31738 54.5429359 54.5416067 0.7615569
63 49.600 80.800 1.29624 55.1857392 55.1843944 0.7705320
64 50.400 79.200 1.27485 55.8177839 55.8164237 0.7793570
65 51.200 77.600 1.25321 56.4389468 56.4375714 0.7880300
66 52.000 76.000 1.23132 57.0491068 57.0477166 0.7965493
67 52.800 74.400 1.20920 57.6481450 57.6467402 0.8049134
68 53.600 72.800 1.18684 58.2359446 58.2345255 0.8131206
69 54.400 71.200 1.16425 58.8123910 58.8109578 0.8211692
70 55.200 69.600 1.14143 59.3773718 59.3759248 0.8290578
71 56.000 68.000 1.11839 59.9307768 59.9293163 0.8367847
72 56.800 66.400 1.09513 60.4724982 60.4710245 0.8443485
73 57.600 64.800 1.07165 61.0024304 61.0009438 0.8517477
74 58.400 63.200 1.04797 61.5204700 61.5189708 0.8589808
75 59.200 61.600 1.02409 62.0265161 62.0250046 0.8660465
76 60.000 60.000 1.00000 62.5204700 62.5189464 0.8729434
77 60.800 58.400 0.97572 63.0022354 63.0007001 0.8796700
78 61.600 56.800 0.95125 63.4717184 63.4701717 0.8862252
79 62.400 55.200 0.92659 63.9288275 63.9272696 0.8926076
80 63.200 53.600 0.90176 64.3734735 64.3719048 0.8988159
81 64.000 52.000 0.87674 64.8055698 64.8039905 0.9048491
82 64.800 50.400 0.85156 65.2250321 65.2234426 0.9107059
83 65.600 48.800 0.82621 65.6317787 65.6301793 0.9163851
84 66.400 47.200 0.80070 66.0257302 66.0241212 0.9218856
85 67.200 45.600 0.77503 66.4068098 66.4051915 0.9272064
86 68.000 44.000 0.74921 66.7749434 66.7733161 0.9323465
87 68.800 42.400 0.72325 67.1300590 67.1284231 0.9373048
88 69.600 40.800 0.69714 67.4720874 67.4704432 0.9420804
89 70.400 39.200 0.67090 67.8009621 67.7993098 0.9466723
90 71.200 37.600 0.64453 68.1166188 68.1149589 0.9510797
91 72.000 36.000 0.61803 68.4189961 68.4173288 0.9553016
92 72.800 34.400 0.59142 68.7080349 68.7063606 0.9593374
93 73.600 32.800 0.56468 68.9836790 68.9819979 0.9631860
94 74.400 31.200 0.53784 69.2458746 69.2441871 0.9668469
95 75.200 29.600 0.51089 69.4945705 69.4928770 0.9703194
96 76.000 28.000 0.48384 69.7297184 69.7280191 0.9736026
97 76.800 26.400 0.45670 69.9512723 69.9495676 0.9766961
98 77.600 24.800 0.42947 70.1591890 70.1574793 0.9795991
99 78.400 23.200 0.40216 70.3534281 70.3517136 0.9823112
100 79.200 21.600 0.37476 70.5339517 70.5322328 0.9848318
101 80.000 20.000 0.34730 70.7007245 70.6990015 0.9871603
102 80.800 18.400 0.31976 70.8537140 70.8519874 0.9892964
103 81.600 16.800 0.29217 70.9928905 70.9911605 0.9912397
104 82.400 15.200 0.26451 71.1182268 71.1164937 0.9929897
105 83.200 13.600 0.23681 71.2296985 71.2279626 0.9945461
106 84.000 12.000 0.20906 71.3272837 71.3255455 0.9959087
107 84.800 10.400 0.18127 71.4109636 71.4092234 0.9970771
108 85.600 8.800 0.15344 71.4807218 71.4789799 0.9980511
109 86.400 7.200 0.12558 71.5365447 71.5348014 0.9988305
110 87.200 5.600 0.09770 71.5784213 71.5766770 0.9994152
111 88.000 4.000 0.06980 71.6063437 71.6045987 0.9998050
112 88.800 2.400 0.04188 71.6203062 71.6185608 1.0000000
[ 71.618560834737543729429789502783231967988796348224, 1.0] <=== 1.0 ratio indicates
113 89.600 0.800 0.01396 71.6203062 71.6185608 1.0000000 exact match
[ 71.618560834737543729429789502783231967988796348224, 1.0] to 50 significant digits
114 90.400 -0.800 0.01396 71.6342687 71.6325230 1.0001950
[ 71.632523015076689001051700890480007430476705061781,
1.0001949519813915768975237841226157112052042164957]
[ 71.632523015076689001051700890480007430476705061781,
1.0001949519813915768975237841226157112052042164957]
>>
Edited on March 11, 2021, 10:01 am
|
|
Posted by Charlie
on 2021-03-11 10:00:29 |
computer solution
