x = linspace(-12,12,500);
y1 = (18-x.^3).^(1/3);
y2=4-x;
figure
plot(x,y1 )
% t=title(['dist is ',num2str(dist) ]);
% t.FontSize=16;
grid
hold on
plot(x,y2 )
axis equal
clc
for x=[1.6 2.4]
y=4-x;
x^5+y^5;
for i=1:200
y=(18-x^3)'^(1/3);
x=4-y;
disp([x y x^5+y^5])
end
disp(' ')
end
The first part provided a graph of both equations, leading to the initial approximation for the second part. One section of the curve, with another intersection with the linear equation, had to be discarded as it showed only the real parts of complex solutions. Matlab warned to look out for that.
1.59537927238128 2.40462072761872 90.7309405481401
1.59334113111879 2.40665886888121 91.0063839368708
1.5924469678766 2.4075530321234 91.1276965440912
1.59205561522047 2.40794438477953 91.1808825205023
1.59188450823372 2.40811549176628 91.2041537827805
1.59180973099444 2.40819026900556 91.2143271077964
1.59177705832246 2.40822294167754 91.2187728049888
1.59176278378331 2.40823721621669 91.220715230139
1.59175654754027 2.40824345245973 91.2215638574633
1.59175382310336 2.40824617689664 91.2219346029582
1.59175263288312 2.40824736711688 91.2220965707659
1.59175211291523 2.40824788708477 91.2221673293092
1.59175188575876 2.40824811424124 91.2221982413663
1.5917517865218 2.4082482134782 91.2222117458016
1.59175174316855 2.40824825683145 91.2222176454309
1.59175172422899 2.40824827577101 91.2222202227781
1.59175171595494 2.40824828404506 91.2222213487333
1.59175171234029 2.40824828765971 91.2222218406247
1.59175171076118 2.40824828923882 91.2222220555153
1.59175171007132 2.40824828992868 91.2222221493936
1.59175170976994 2.40824829023006 91.2222221904059
1.59175170963828 2.40824829036172 91.2222222083227
1.59175170958076 2.40824829041924 91.22222221615
1.59175170955563 2.40824829044437 91.2222222195695
1.59175170954465 2.40824829045535 91.2222222210633
1.59175170953986 2.40824829046014 91.2222222217159
1.59175170953776 2.40824829046224 91.2222222220011
1.59175170953685 2.40824829046315 91.2222222221255
1.59175170953645 2.40824829046355 91.22222222218
1.59175170953627 2.40824829046373 91.2222222222037
1.5917517095362 2.4082482904638 91.2222222222141
1.59175170953616 2.40824829046384 91.2222222222187
1.59175170953615 2.40824829046385 91.2222222222207
1.59175170953614 2.40824829046386 91.2222222222215
1.59175170953614 2.40824829046386 91.2222222222219
1.59175170953614 2.40824829046386 91.2222222222221
1.59175170953614 2.40824829046386 91.2222222222221
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
2.38965042376012 1.61034957623988 88.7534239459198
2.36707907439639 1.63292092560361 85.9226628303019
2.32053470285017 1.67946529714983 80.649919725581
2.23437705130717 1.76562294869283 72.8495681643549
2.10129313938261 1.89870686061739 65.6437534889146
1.94156706125099 2.05843293874901 64.5465384966203
1.79773591699478 2.20226408300522 70.5791952883184
1.69855294478804 2.30144705521196 78.704400662285
1.64267951746184 2.35732048253816 84.7545014207008
1.61494236533576 2.38505763466424 88.1627773149673
1.60207542957028 2.39792457042972 89.8364904174002
1.59629968859119 2.40370031140881 90.6070407407662
1.5937458995693 2.4062541004307 90.9515628957968
1.59262431117346 2.40737568882654 91.1036132163538
1.59213318886325 2.40786681113675 91.1703356541285
1.59191841632228 2.40808158367772 91.1995413043284
1.59182454787397 2.40817545212603 91.2123111348853
1.59178353199745 2.40821646800255 91.2178919160256
1.59176561204195 2.40823438795805 91.2203303656022
1.59175778313496 2.40824221686504 91.2213957167498
1.59175436289729 2.40824563710271 91.2218611467711
1.59175286870159 2.40824713129841 91.2220644800262
1.59175221593645 2.40824778406355 91.2221533099145
1.59175193076525 2.40824806923475 91.2221921167624
1.5917518061836 2.4082481938164 91.2222090701695
1.59175175175812 2.40824824824188 91.2222164765382
1.59175172798148 2.40824827201852 91.2222197121287
1.59175171759428 2.40824828240572 91.222221125648
1.59175171305646 2.40824828694354 91.2222217431663
1.59175171107405 2.40824828892595 91.222222012939
1.591751710208 2.408248289792 91.2222221307935
1.59175170982965 2.40824829017035 91.2222221822801
1.59175170966436 2.40824829033564 91.2222222047728
1.59175170959215 2.40824829040785 91.2222222145992
1.59175170956061 2.40824829043939 91.222222218892
1.59175170954683 2.40824829045317 91.2222222207673
1.59175170954081 2.40824829045919 91.2222222215866
1.59175170953818 2.40824829046182 91.2222222219445
1.59175170953703 2.40824829046297 91.2222222221009
1.59175170953653 2.40824829046347 91.2222222221692
1.59175170953631 2.40824829046369 91.222222222199
1.59175170953621 2.40824829046379 91.2222222222121
1.59175170953617 2.40824829046383 91.2222222222178
1.59175170953615 2.40824829046385 91.2222222222202
1.59175170953614 2.40824829046386 91.2222222222213
1.59175170953614 2.40824829046386 91.2222222222218
1.59175170953614 2.40824829046386 91.222222222222
1.59175170953614 2.40824829046386 91.2222222222221
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
1.59175170953614 2.40824829046386 91.2222222222222
Even the second approximation gravitated to the first solution, but by symmetry, we can see the x and y values are interchangeable. In either case, the sum of the 5th powers sought is 91.2222222222222 or 91 and 2/9.
x and y values:
2 - 1/sqrt(6) 2 + 1/sqrt(6)
1.59175170953614 2.40824829046386
solution for sum of fifth powers:
91 + 2/9
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Posted by Charlie
on 2023-11-26 09:15:18 |
computer solution
