Alex owns one of those folding rulers where each segment is exactly 1 foot long. While playing with the open ruler he formed it into a triangle. Then he refolded it into a second triangle with double the area.
What is the smallest possible length of the ruler? What if the second triangle has 3 times the area?
The ruler folds upon itself so that smaller periphery triangles can be used with those of longer total ruler length.
clearvars,clc
tri=double.empty(0,3);
area=[];
for tot=3:17
for a=1:tot
for b=1:tot
c=tot-a-b;
s=sum([a b c])/2;
A =sqrt(s*(s-a)*(s-b)*(s-c));
if isreal(A) && A>0 && c>0
area(end+1)=A;
tri(end+1,:)=[a b c];
end
end
end
end
for a=1:length(tri)
for b=1:length(tri)
if area(b)==2*area(a)
fprintf('%19.15f %19.15f %3d %3d %3d %3d %3d %3d %4d\n', ...
[area(a) area(b) tri(a,:) tri(b,:) max([sum(tri(a,:)) sum(tri(b,:))])])
end
end
end
disp(' ')
for a=1:length(tri)
for b=1:length(tri)
if area(b)==3*area(a)
fprintf('%19.15f %19.15f %3d %3d %3d %3d %3d %3d %4d\n', ...
[area(a) area(b) tri(a,:) tri(b,:) max([sum(tri(a,:)) sum(tri(b,:))])])
end
end
end
finds
4.472135954999580 8.944271909999159 3 3 4 3 6 7 16
4.472135954999580 8.944271909999159 3 3 4 3 7 6 16
4.472135954999580 8.944271909999159 3 3 4 6 3 7 16
4.472135954999580 8.944271909999159 3 3 4 6 7 3 16
4.472135954999580 8.944271909999159 3 3 4 7 3 6 16
4.472135954999580 8.944271909999159 3 3 4 7 6 3 16
4.472135954999580 8.944271909999159 3 4 3 3 6 7 16
4.472135954999580 8.944271909999159 3 4 3 3 7 6 16
4.472135954999580 8.944271909999159 3 4 3 6 3 7 16
4.472135954999580 8.944271909999159 3 4 3 6 7 3 16
4.472135954999580 8.944271909999159 3 4 3 7 3 6 16
4.472135954999580 8.944271909999159 3 4 3 7 6 3 16
4.472135954999580 8.944271909999159 4 3 3 3 6 7 16
4.472135954999580 8.944271909999159 4 3 3 3 7 6 16
4.472135954999580 8.944271909999159 4 3 3 6 3 7 16
4.472135954999580 8.944271909999159 4 3 3 6 7 3 16
4.472135954999580 8.944271909999159 4 3 3 7 3 6 16
4.472135954999580 8.944271909999159 4 3 3 7 6 3 16
4.898979485566356 9.797958971132712 2 5 5 4 5 7 16
4.898979485566356 9.797958971132712 2 5 5 4 7 5 16
4.898979485566356 9.797958971132712 2 5 5 5 4 7 16
4.898979485566356 9.797958971132712 2 5 5 5 7 4 16
4.898979485566356 9.797958971132712 2 5 5 7 4 5 16
4.898979485566356 9.797958971132712 2 5 5 7 5 4 16
6.000000000000000 12.000000000000000 3 4 5 5 5 6 16
6.000000000000000 12.000000000000000 3 4 5 5 6 5 16
6.000000000000000 12.000000000000000 3 4 5 6 5 5 16
6.000000000000000 12.000000000000000 3 5 4 5 5 6 16
6.000000000000000 12.000000000000000 3 5 4 5 6 5 16
6.000000000000000 12.000000000000000 3 5 4 6 5 5 16
6.000000000000000 12.000000000000000 4 3 5 5 5 6 16
6.000000000000000 12.000000000000000 4 3 5 5 6 5 16
6.000000000000000 12.000000000000000 4 3 5 6 5 5 16
6.000000000000000 12.000000000000000 4 5 3 5 5 6 16
6.000000000000000 12.000000000000000 4 5 3 5 6 5 16
6.000000000000000 12.000000000000000 4 5 3 6 5 5 16
4.898979485566356 9.797958971132712 5 2 5 4 5 7 16
4.898979485566356 9.797958971132712 5 2 5 4 7 5 16
4.898979485566356 9.797958971132712 5 2 5 5 4 7 16
4.898979485566356 9.797958971132712 5 2 5 5 7 4 16
4.898979485566356 9.797958971132712 5 2 5 7 4 5 16
4.898979485566356 9.797958971132712 5 2 5 7 5 4 16
6.000000000000000 12.000000000000000 5 3 4 5 5 6 16
6.000000000000000 12.000000000000000 5 3 4 5 6 5 16
6.000000000000000 12.000000000000000 5 3 4 6 5 5 16
6.000000000000000 12.000000000000000 5 4 3 5 5 6 16
6.000000000000000 12.000000000000000 5 4 3 5 6 5 16
6.000000000000000 12.000000000000000 5 4 3 6 5 5 16
4.898979485566356 9.797958971132712 5 5 2 4 5 7 16
4.898979485566356 9.797958971132712 5 5 2 4 7 5 16
4.898979485566356 9.797958971132712 5 5 2 5 4 7 16
4.898979485566356 9.797958971132712 5 5 2 5 7 4 16
4.898979485566356 9.797958971132712 5 5 2 7 4 5 16
4.898979485566356 9.797958971132712 5 5 2 7 5 4 16
2.904737509655563 8.714212528966687 2 3 4 3 6 6 15
2.904737509655563 8.714212528966687 2 3 4 6 3 6 15
2.904737509655563 8.714212528966687 2 3 4 6 6 3 15
2.904737509655563 8.714212528966687 2 4 3 3 6 6 15
2.904737509655563 8.714212528966687 2 4 3 6 3 6 15
2.904737509655563 8.714212528966687 2 4 3 6 6 3 15
2.904737509655563 8.714212528966687 3 2 4 3 6 6 15
2.904737509655563 8.714212528966687 3 2 4 6 3 6 15
2.904737509655563 8.714212528966687 3 2 4 6 6 3 15
2.904737509655563 8.714212528966687 3 4 2 3 6 6 15
2.904737509655563 8.714212528966687 3 4 2 6 3 6 15
2.904737509655563 8.714212528966687 3 4 2 6 6 3 15
2.904737509655563 8.714212528966687 4 2 3 3 6 6 15
2.904737509655563 8.714212528966687 4 2 3 6 3 6 15
2.904737509655563 8.714212528966687 4 2 3 6 6 3 15
2.904737509655563 8.714212528966687 4 3 2 3 6 6 15
2.904737509655563 8.714212528966687 4 3 2 6 3 6 15
2.904737509655563 8.714212528966687 4 3 2 6 6 3 15
so that a 16-foot ruler can do the double-area, and a 15-foot ruler can do the triple:
3-3-4 and 3-6-7 for areas 4.472135954999580 and 8.944271909999159.
2-3-4 and 3-6-6 for areas 2.904737509655563 and 8.714212528966687.
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Posted by Charlie
on 2024-06-28 10:56:22 |
computer solution
