NMMRUS 146 Loesung

Welche Strecke bewältigt der Kurier?

zurück zur Aufgabenstellung

Ich rechne die Aufgabe lieber "allgemein" als mit Zahlen - Zahlen kann man am Schluss immer noch einsetzen. So muss man weniger schreiben (außer diesem Absatz) uns sieht die Zusammenhänge besser.

Die "Länge" der Armee bzw. die Seitenlänge des Quadrats ist 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 a} . Wir wissen schon, dass $a=50Mi$. Die Geschwindigkeit des Kuriers ist $v_{K}$. Die Gewschwindigkeit der Armee ist $v_{A}$. Die Stecke, die der Kurier in den einzelnen Abschnitten zurücklegt nenne ich $x_{i}$ die Zeit, die er für diesen Abschnitt braucht ist $t_{i}$.

einfache (eindimensionale) Variante

Der Kurier reitet im ersten Abschnitt die Strecke $x_{1}$ - während die Armee in der gleiche Zeit die Strcke $(x_{1}-a)$ zurücklegt.

$x_{1}=v_{K}\cdot t_{1}$
$x_{1}-a=v_{A}\cdot t_{1}$

Durch "Umformen" entledigen wir uns dem $t_{1}$ und finden das $x_{1}$.

$t_{1}={x_{1} \over v_{K}}$
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_1-a=v_A\cdot {x_1\over v_K}}
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_1\cdot (1-{v_A\over v_K})=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_1={a\over{1-{v_A\over v_K}}}}

Im Zweiten Abschnitt reitet der Kurier die Strecke 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_2} - während die Armee die Stecke 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 (a-x_2)} zurücklegt.

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_2=v_K\cdot t_2}
$a-x_{2}=v_{A}\cdot t_{2}$

Wieder arbeiten wir das 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_2} heraus um das 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_2} auszudrücken.

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_2={x_2\over v_K}}
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 a-x_2=v_A\cdot{x_2\over v_K}}
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_2\cdot(1+{v_A\over v_K})=a}
$x_{2}={a \over {1+{v_{A} \over v_{K}}}}$

Die Antwort aud die Frage ist 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=x_1+x_2} .

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={a\over{1-{v_A\over v_K}}} + {a\over{1+{v_A\over v_K}}}}

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={{a\cdot(1+{v_A\over v_K}) + a\cdot(1-{v_A\over v_K})}\over{(1-{v_A\over v_K})(1+{v_A\over v_K})}}}

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={{2a}\over{1-({v_A\over v_K})^2}}}

Was uns noch zur endgültigen Beantwortung fehlt ist 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} und $v_{K}$. Dazu hilt uns ein Umstand, den wir noch nicht verwendet haben: Die Armee legt während dem Hin- und Herreiten genau die Strecke 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 a} zurück. Die exakten Geschwindigkeiten 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} und 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_K} sind uninteressant bzw. aus der Aufgabenstellung nicht zu ermitteln. Das einzige worauf es ankommt ist der Quotient aus den Geschwindigkeiten - um wieviel die Armee langsamer vorankommt als der Kurier.

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 q={v_A\over v_K}}

$x={{2a} \over {1-q^{2}}}$

Dieser Quotient ist auch genau der Faktor um den die zurückgelegte Strecke der Armee kleiner ist als die des Kuriers. Die Strecke der Armee ist die Strecke des Kuriers mal q. Weiters ist die Strecke der Armee gleich ihre Länge:

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 a=x\cdot q}

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 a={{2aq}\over{1-q^2}}}

Da sieht man jetzt auch schön, dass die Länge der Armee für die Berechnung des Quotienten irrelevant ist, denn man kann durch a kürzen:

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 1={{2q}\over{1-q^2}}}

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 1-q^2=2q}
$q^{2}+2q-1=0$
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 q_{1,2}=-1\pm \sqrt{1+1}}

Der Quotient ist ganz sicher positiv, darum gibt es nur eine Lösung:

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 q=\sqrt{2}-1}

Somit ist die Antwort auf die Frage "Welchen Weg legt der Kurier zurück" :

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={{2a}\over{1-(\sqrt{2}-1)^2}}}
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={{2a}\over{1-(2-2\sqrt{2}+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 x={{2a}\over{-2+2\sqrt{2}}}}
$x={{a} \over {{\sqrt {2}}-1}}$
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 x={{a\cdot (\sqrt{2}+1)}\over{(\sqrt{2}-1)\cdot({\sqrt{2}+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 x={{a\cdot (\sqrt{2}+1)}\over{2-1}}}
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 x=a\cdot (\sqrt{2}+1)}

Das heißt, dass der Kurier eine Strecke zurücklegt, die aus einer Seitenlänge plus der Diagonale des Quadrates mit der Seitenlänge a entspricht.

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 \approx 120.71 Mi}