A regular tetrahedron with an edge of 30cm rests on one of its faces. Assuming it is hollow, 2 liters of water are poured into it. Find the height and the area of the free surface of water.
V = h*(a+sqrt(a*b)+b)/3 = 2000 (volume of frustum)
where a and b are the areas of the top and the bottom
b = sqrt(3)*30^2 / 4
H = sqrt(2/3) * 30 (height of full pyramid)
a = ((H - h) / H)^2 * b (inverse square)
A program,
V=2000;
b = sqrt(3)*30^2 / 4;
H = sqrt(2/3) * 30;
h=.3;
for i=1:35;
a=((H - h) / H)^2 * b;
h=3*V/(a+sqrt(a*b)+b)
end
gives successively better approximations and settles down to h = 6.88676784410156 cm. The area of the top triangle of the frustum is a = 201.38093198126 cm^2 and of the bottom is b = 389.711431702997 cm^2, so this seems reasonable.
The top surface is a little over 28% the total height of the pyramid, which is H = 24.4948974278318.
The successive approximations:
a =
380.223946871734
h =
5.19537268466759
a =
241.927436767379
h =
6.3918700396322
a =
212.860124589897
h =
6.73711474134795
a =
204.818588538331
h =
6.84107912175356
a =
202.427354815388
h =
6.87277889353065
a =
201.701037183741
h =
6.8824809430678
a =
201.479000911471
h =
6.8854537732732
a =
201.410990674467
h =
6.88636500655585
a =
201.390146444157
h =
6.88664434832726
a =
201.383756787975
h =
6.88672998436198
a =
201.381797971915
h =
6.88675623752395
a =
201.381197465925
h =
6.88676428589583
a =
201.381013370354
h =
6.88676675326917
a =
201.380956932559
h =
6.88676750968712
a =
201.380939630533
h =
6.88676774158075
a =
201.380934326283
h =
6.88676781267195
a =
201.38093270017
h =
6.88676783446624
a =
201.380932201655
h =
6.88676784114768
a =
201.380932048826
h =
6.88676784319599
a =
201.380932001974
h =
6.88676784382394
a =
201.38093198761
h =
6.88676784401645
a =
201.380931983207
h =
6.88676784407547
a =
201.380931981857
h =
6.88676784409356
a =
201.380931981443
h =
6.88676784409911
a =
201.380931981316
h =
6.88676784410081
a =
201.380931981277
h =
6.88676784410133
a =
201.380931981265
h =
6.88676784410149
a =
201.380931981262
h =
6.88676784410154
a =
201.380931981261
h =
6.88676784410155
a =
201.38093198126
h =
6.88676784410156
a =
201.38093198126
h =
6.88676784410156
a =
201.38093198126
h =
6.88676784410156
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Posted by Charlie
on 2024-06-10 09:24:14 |
computer-aided solution
