Difference between revisions of "NMMRUS 123 Loesung"
| Line 40: | Line 40: | ||
<math>x={{L(f-a)(b+f)}\over {(b+f)(f-a)-2f(b-f)}}</math><br/> | <math>x={{L(f-a)(b+f)}\over {(b+f)(f-a)-2f(b-f)}}</math><br/> | ||
<math>x={{40(1.5-10)(15+1.5)}\over {(15+1.5)(1.5-10)-2\cdot 1.5\cdot (15 - 1.5)}}</math><br/> | <math>x={{40(1.5-10)(15+1.5)}\over {(15+1.5)(1.5-10)-2\cdot 1.5\cdot (15 - 1.5)}}</math><br/> | ||
| − | <math>x=31.037344</math>M<br/> | + | <math>x=31.037344</math> M<br/> |
<math>t=f x + a(L-x)=f x + aL -ax=x(f-a)+aL</math><br/> | <math>t=f x + a(L-x)=f x + aL -ax=x(f-a)+aL</math><br/> | ||
| − | <math>t=136.182573</math>min<br/> | + | <math>t=136.182573</math> min<br/> |
| + | |||
| + | D.h. Die ganze Aktion dauert 136.18 Minuten. | ||
Revision as of 00:05, 2 January 2009
Das Tandem
Wie verfahren die Drei? Zu Fuß ist A am schnellsten - er solltete die länste Strecke zurücklegen, B die andere Strecke und C sollte nie gehen und immer am Rad fahren.
Am Besten beginnt B zu maschieren, während A+C mit dem Tandem losdüsen. Am Punkt X wird A von C abgesetzt und maschiert Richtung Ziel. C fährt alleine mit dem Rad zurück um B abzuholen bei Y hat C B erreicht - beide radeln jetzt Richtung Ziel. X wurde so gewählt, dass A, B+C gleichzeitig eintreffen. Die Konstilation ist so gewählt, dass B eine kürzere Strecke zurücklegen muss wie A. Weiters sind die ganze Zeit alle 3 "beschäftigt" => es gibt keine Totzeiten => das ist die optimale Lösung.
Problem: Wo ist X - wo ist Y - und wie lange dauert das alles? Los geht's:
B marschiert Richtung Y während A+C losradeln, bei X wird A abgesetzt und C radelt wieder zurück zu Y.
Laut Angabe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_a={1 \over 10}} M/min; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_b={1 \over 15}} M/min; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_c={1 \over 20}} M/min; M/min; L=40M.
Da wir ab nun mit "Zeiten" rechnen (Zeit = Weg / Geschwindigkeit), will ich jetzt die Reziprokwerte einführen und diese (?) Zeitikeiten nennen: a=10min/M; b=15min/M; c=20min/M; f= 1.5min/M.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1=y b}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1=(2x-y)f}
Damit lässt sich y lösen:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y b = (2x-y) f }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y b = 2xf - y f}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(b+f)=2xf}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y={{2xf} \over {b+f}}}
Ab nun ist y kein Thema mehr, da wir es mittels x ausdrücken können. Ab nun suchen wir x, dass so gewählt wird, dass A genausolange maschiert wie, C braucht um B abzuholen und zum Ziel zu gelangen...
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=f x + a (L-x)}
[1]
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=t_1+(L-y) f}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t={{2xf} \over {b+f}} b + (L - {{2xf} \over {b+f}}) f }
[2]
Jetzt wird [1] und [2] zusammengeführt:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f x + a L - a x = x {{2fb}\over {b+f}} + L f - x {{2xf^2}\over {b+f}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(f-a-{{2fb}\over {b+f}}+{{2f^2}\over {b+f}})=L f - a L}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x{{(b+f)(f-a)-2fb+2f^2}\over {b+f}} = L(f-a)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x={{L(f-a)(b+f)}\over {(b+f)(f-a)-2f(b-f)}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x={{40(1.5-10)(15+1.5)}\over {(15+1.5)(1.5-10)-2\cdot 1.5\cdot (15 - 1.5)}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=31.037344}
M
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=f x + a(L-x)=f x + aL -ax=x(f-a)+aL}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=136.182573}
min
D.h. Die ganze Aktion dauert 136.18 Minuten.