I cut off one corner of an empty box to form an open 'pocket', so that each of the length a, breadth b, and depth c, of the initial corner was a different whole number of centimetres, and the area of each face of the 'pocket' was also an integer.
What is the area of the open 4th side of the 'pocket', in terms of a,b, and c?
It's hard to say what's wanted here.
I wrote a program that finds the area of the open face based on using the pythagorean theorem to find the lengths of the sides of the open triangular face, and then Heron's formula for finding the area of the triangle.
The only restriction given is basically that at least two out of a, b and c must be even.
c' = sqrt(a^2 + b^2)
b' = sqrt(a^2 + c^2)
a' = sqrt(c^2 + b^2)
s = (a' + b' + c')/2
area = sqrt(s * (s - a') * (s - b') * (s - c'))
Is what's wanted more than just appropriately substituting a, b and c within the last formula?
Or is there a simpler formula looked for, perhaps based on those situations were the 4th, open, side also has integral area, shown among the below examples.
Some examples:
a b c area a+b+c
1 2 4 4.5825756950 7
1 2 6 6.7823299831 9
1 4 4 8.4852813742 9
2 3 4 7.8102496759 9
2 4 4 9.7979589711 10
1 2 8 9.0000000000 11
1 4 6 12.5299640861 11
2 3 6 11.2249721603 11
2 4 5 11.8743420870 11
3 4 4 11.6619037897 11
2 4 6 14.0000000000 12
1 2 10 11.2249721603 13
1 4 8 16.6132477258 13
1 6 6 18.4932420089 13
2 3 8 14.7309198627 13
2 4 7 16.1554944214 13
2 5 6 16.9115345253 13
3 4 6 16.1554944214 13
2 4 8 18.3303027798 14
2 6 6 19.8997487421 14
1 2 12 13.4536240471 15
1 4 10 20.7123151772 15
1 6 8 24.5153013443 15
2 3 10 18.2756668825 15
2 4 9 20.5182845287 15
2 5 8 22.1133443875 15
2 6 7 22.9346898824 15
3 4 8 20.8806130178 15
3 6 6 22.0454076850 15
4 5 6 21.6564078277 15
2 4 10 22.7156333832 16
2 6 8 26.0000000000 16
4 6 6 24.7386337537 16
1 2 14 15.6843871414 17
1 4 12 24.8193472920 17
1 6 10 30.5614135799 17
1 8 8 32.4961536185 17
2 3 12 21.8403296678 17
2 4 11 24.9198715888 17
2 5 10 27.3861278753 17
2 6 9 29.0860791445 17
2 7 8 29.9499582637 17
3 4 10 25.7099202644 17
3 6 8 28.3019433962 17
4 5 8 27.4954541697 17
4 6 7 27.9463772250 17
5 6 6 27.8208554865 17
2 4 12 27.1293199325 18
2 6 10 32.1869538789 18
2 8 8 33.9411254970 18
4 6 8 31.2409987036 18
1 2 16 17.9164728672 19
1 4 14 28.9309522830 19
1 6 12 36.6196668472 19
1 8 10 40.5092582011 19
2 3 14 25.4165300543 19
2 4 13 29.3428015022 19
2 5 12 32.6955654485 19
2 6 11 35.2987251894 19
2 7 10 37.0675059857 19
2 8 9 37.9605057922 19
3 4 12 30.5941170816 19
3 6 10 34.7275107084 19
3 8 8 36.2215405525 19
4 5 10 33.5410196625 19
4 6 9 34.5976877840 19
4 7 8 35.1567916625 19
5 6 8 34.6554469023 19
2 4 14 31.5594676761 20
2 6 12 38.4187454246 20
2 8 10 42.0000000000 20
4 6 10 38.0000000000 20
4 8 8 39.1918358845 20
1 2 18 20.1494416796 21
1 4 16 33.0454232837 21
1 6 14 42.6848919408 21
1 8 12 48.5386443980 21
1 10 10 50.4975246918 21
2 3 16 29.0000000000 21
2 4 15 33.7786915081 21
2 5 14 38.0263066837 21
2 6 13 41.5451561557 21
2 7 12 44.2379927212 21
2 8 11 46.0543157587 21
2 9 10 46.9680742633 21
3 4 14 35.5105618091 21
3 6 12 41.2431812546 21
3 8 10 44.3734154647 21
4 5 12 39.6988664826 21
4 6 11 41.4366987102 21
4 7 10 42.6731765867 21
4 8 9 43.3128156554 21
5 6 10 41.8330013267 21
5 8 8 42.7083130081 21
6 7 8 42.4381903478 21
2 4 16 36.0000000000 22
2 6 14 44.6766158074 22
2 8 12 50.1198563446 22
2 10 10 51.9615242271 22
4 6 12 44.8998886413 22
4 8 10 47.4973683482 22
6 8 8 46.6476151588 22
1 2 20 22.3830292856 23
1 4 18 37.1618083521 23
1 6 16 48.7544869730 23
1 8 14 56.5773806393 23
1 10 12 60.5061980296 23
2 3 18 32.5883414736 23
2 4 17 38.2230297072 23
2 5 16 43.3704968844 23
2 6 15 47.8121323515 23
2 7 14 51.4392845984 23
2 8 13 54.1940956194 23
2 9 12 56.0446250768 23
2 10 11 56.9736781330 23
3 4 16 40.4474968323 23
3 6 14 47.8121323515 23
3 8 12 52.6497863244 23
3 10 10 54.3139024560 23
4 5 14 45.9238500128 23
4 6 13 48.3838816136 23
4 7 12 50.3587132481 23
4 8 11 51.7300686255 23
4 9 10 52.4309069157 23
5 6 12 49.2036584006 23
5 8 10 51.2347538298 23
6 7 10 50.6557005677 23
6 8 9 51.0000000000 23
7 8 8 50.9116882454 23
2 4 18 40.4474968323 24
2 6 16 50.9509568114 24
2 8 14 58.2752091373 24
2 10 12 62.0000000000 24
4 6 14 51.8844870843 24
4 8 12 56.0000000000 24
4 10 10 57.4456264654 24
6 8 10 55.4616984954 24
1 2 22 24.6170672502 25
1 4 20 41.2795348811 25
1 6 18 54.8270006475 25
1 8 16 64.6219776856 25
1 10 14 70.5265907300 25
1 12 12 72.4982758416 25
2 3 20 36.1801050303 25
2 4 19 42.6731765867 25
2 5 18 48.7237108603 25
2 6 17 54.0925133452 25
2 7 16 58.6600375042 25
2 8 15 62.3618473107 25
2 9 14 65.1613382306 25
2 10 13 67.0373030484 25
2 11 12 67.9779375974 25
3 4 18 45.3982378513 25
3 6 16 54.4150714417 25
3 8 14 61.0000000000 25
3 10 12 64.4127316608 25
4 5 16 52.1919534028 25
4 6 15 55.3985559379 25
4 7 14 58.1463670404 25
4 8 13 60.2992537267 25
4 9 12 61.7737808459 25
4 10 11 62.5219961294 25
5 6 14 56.6921511322 25
5 8 12 60.0333240792 25
5 10 10 61.2372435696 25
6 7 12 59.1692487699 25
6 8 11 60.0083327547 25
6 9 10 60.4483250388 25
7 8 10 60.0749531835 25
2 4 20 44.8998886413 26
2 6 18 57.2363520850 26
2 8 16 66.4529909033 26
2 10 14 72.0832851638 26
2 12 12 73.9729680356 26
4 6 16 58.9236794506 26
4 8 14 64.6219776856 26
4 10 12 67.6461381012 26
6 8 12 64.6219776856 26
6 10 10 65.5743852430 26
1 2 24 26.8514431642 27
1 4 22 45.3982378513 27
1 6 20 60.9015599143 27
1 8 18 72.6704891961 27
1 10 16 80.5543294926 27
1 12 14 84.5044377533 27
2 3 22 39.7743636027 27
2 4 21 47.1274866718 27
2 5 20 54.0832691320 27
2 6 19 60.3821165578 27
2 7 18 65.8938540381 27
2 8 17 70.5478560978 27
2 9 16 74.3034319530 27
2 10 15 77.1362431027 27
2 11 14 79.0316392339 27
2 12 13 79.9812478022 27
3 4 20 50.3587132481 27
3 6 18 61.0409698481 27
3 8 16 69.3974062916 27
3 10 14 74.6056298144 27
3 12 12 76.3675323681 27
4 5 18 58.4893152636 27
4 6 17 62.4579858785 27
4 7 16 66.0000000000 27
4 8 15 68.9637585983 27
4 9 14 71.2530701093 27
4 10 13 72.8079665971 27
4 11 12 73.5934779719 27
5 6 16 64.2572953057 27
5 8 14 69.0000000000 27
5 10 12 71.5891053164 27
6 7 14 67.8675180038 27
6 8 13 69.2892488053 27
6 9 12 70.2922470832 27
6 10 11 70.8096038684 27
7 8 12 69.6562990691 27
7 10 10 70.3562363974 27
8 9 10 70.1498396292 27
2 4 22 49.3558507170 28
2 6 20 63.5295206971 28
2 8 18 74.6458304261 28
2 10 16 82.1948903521 28
2 12 14 86.0000000000 28
4 6 18 66.0000000000 28
4 8 16 73.3212111193 28
4 10 14 78.0000000000 28
4 12 12 79.5989949685 28
6 8 14 74.0000000000 28
6 10 12 76.1314652427 28
8 10 10 75.4983443527 28
1 2 26 29.0860791445 29
1 4 24 49.5176736126 29
1 6 22 66.9776081986 29
1 8 20 80.7217442824 29
1 10 18 90.5869747811 29
1 12 16 96.5194280961 29
1 14 14 98.4987309563 29
2 3 24 43.3704968844 29
2 4 23 51.5848815061 29
2 5 22 59.4474557908 29
2 6 21 66.6783323127 29
2 7 20 73.1368580129 29
2 8 19 78.7464284904 29
2 9 18 83.4625664595 29
2 10 17 87.2582374335 29
2 11 16 90.1165911472 29
2 12 15 92.0271699011 29
2 13 14 92.9838695689 29
3 4 22 55.3263047745 29
3 6 20 67.6830850361 29
3 8 18 77.8267306264 29
3 10 16 84.8587060943 29
3 12 14 88.4364178379 29
4 5 20 64.8074069841 29
4 6 19 69.5485441976 29
4 7 18 73.8985791474 29
4 8 17 77.6916983982 29
4 9 16 80.8207894047 29
4 10 15 83.2165848855 29
4 11 14 84.8351342310 29
4 12 13 85.6504524215 29
5 6 18 71.8748913043 29
5 8 16 78.0768851838 29
5 10 14 82.1583836258 29
5 12 12 83.5703296631 29
6 7 16 76.6876782802 29
6 8 15 78.7464284904 29
6 9 14 80.3865660418 29
6 10 13 81.5230028888 29
6 11 12 82.1035931004 29
7 8 14 79.5047168412 29
7 10 12 81.1726554943 29
8 9 12 80.7217442824 29
8 10 11 81.0000000000 29
9 10 10 80.9320702812 29
2 4 24 53.8144961883 30
2 6 22 69.8283610004 30
2 8 20 82.8492607088 30
2 10 18 92.3255111007 30
2 12 16 98.0612053771 30
2 14 14 99.9799979996 30
4 6 20 73.1026675300 30
4 8 18 82.0731381147 30
4 10 16 88.4533775500 30
4 12 14 91.7387595294 30
6 8 16 83.5224520713 30
6 10 14 86.9712596206 30
6 12 12 88.1816307402 30
8 10 12 86.6256313108 30
|
|
Posted by Charlie
on 2014-08-14 16:34:12 |
what is wanted?
