NMMRUS 63 Loesung
Wie weit ist es bis Piketown?
Sei s die gesamte Strecke vom Hotel bis Piketown, a die Strecke vom Hotel bis zur Station und b die Strecke von der Station bis Piketown. Alle Strecken werden in Meilen gemessen. Sei 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_f} 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} die Geschwindikeit des Fußgängers bzw. der Kutsche gemessen in Meilen pro Minute (da die Zeitangaben auch in Minuten sind).
Von der 3.Möglichkeit wissen wir, dass der Fußgänger 4 Meilen zurückgelegt hat, wenn die Kutsche in der Station eintrifft.
[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 {a \over v_k} = {4 \over v_f}}
Daraus lassen sich unmittelbar zwei Beziehungen ableiten (wir werden sie später noch brauchen).
[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 {v_f \over v_k} = {4 \over a}}
[3] 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 = v_f {a \over 4}}
Weiters wissen von der 4. Möglichkeit, dass der Fußgänger genau dann in der Station eintrifft, wenn die Pause von 30 Minuten vorbei ist.
[4] 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 - 4} \over v_f} = 30}
[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 v_f = {{a - 4} \over 30}}
Aus [3] und [5] ergibt sich.
[6] 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 = {{(a - 4)} \over 30} \cdot {a \over 4}}
Von Möglichkeit 2 wissen wir, dass die Kutsche den Fußgänger um 1 Meile schlägt. Die Kutsche gibt dem Fußgänger 30 Minuten mehr Zeit, da sie selber solange in der Station verweilt.
[7] 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 = {s \over v_k} + 30}
[8] 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_f \cdot t_2 = s - 1}
[9] 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 s - 1 = {v_f \over v_k} \cdot s + 30 \cdot v_f}
Zusammenfassen durch Herausheben von s.
[10] 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 s \cdot (1 - {v_f \over v_k}) = 30 \cdot v_f + 1}
[11] 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 s = {{30 \cdot v_f + 1} \over {1 - {v_f \over v_k}}}}
Jetzt setzen wir das was wir über die Geschwindikeiten bzw. deren Verhältnis wissen aus [5] und [2] ein.
[12] 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 s = {{30 \cdot {{a - 4} \over 30} + 1} \over {1 - {4 \over a}}}}
[13] 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 s = {{(a - 4) + 1} \over {{a - 4} \over a}}}
[14] 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 s = a \cdot {{a - 3} \over {a - 4}}}
Von der Möglichkeit 4 wissen wir, dass in diesem Fall der Fußgänger die Kutsche um 15 Minuten schlägt. 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_4} ist die Zeit die der Fußgänger für die zweite Teilstrecke b braucht. Die Kutsche hat einerseits 30 Minuten weniger zur Verfügung (wegen der Pause) und braucht dann immer noch 15 Minuten länger.
[15] 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_4 = {b \over v_f}}
[16] 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 {b \over v_k} = t_4 - 30 + 15}
[17] 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 {b \over v_k} = {b \over v_f} - 15}
[18] 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 b \cdot ({1 \over v_f} - {1 \over v_k}) = 15}
[19] 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 b \cdot {{v_k - v_f} \over {v_f \cdot v_k}} = 15}
[20] 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 b = 15 \cdot {{v_k - v_f} \over {v_f \cdot v_k}}}
Einsetzen aus [5] und [6]
[21] 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 b = 15 \cdot { { ( { {a - 4} \over 30 } )^2 \cdot {a \over 4} } \over { ( { {a - 4} \over 30 } ) \cdot {a \over 4} - { {a - 4} \over 30 } } } }
Da kann man durch 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 - 4} \over 30} kürzen.
[22]
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 b = 15 \cdot { { ( { {a - 4} \over 30 } ) \cdot {a \over 4} } \over { { a \over 4 } - 1 } } }
[23] 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 b = { { {15 \cdot (a - 4) \cdot a} \over {30 \cdot 4} } \over { {a - 4} \over 4 } } }
Doppelbrüche auflösen.
[24] 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 b = { { 15 \cdot (a - 4) \cdot a \cdot 4 } \over { 30 \cdot 4 \cdot (a - 4) } } }
Was geht kürzen.
[25]
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 b = { { 15 \cdot a } \over { 30 } } }
[26] 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 b = { a \over 2 }}
Bis jetzt wurde es noch nicht erwähnt aber a + b = s . Da setzen wir jetzt [14] und [26] ein.
[27] 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 + {a \over 2} = a \cdot {{a - 3} \over {a - 4}}}
Mal 2, mal (a - 4)
[28] 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 2 \cdot a \cdot (a - 4) + a \cdot (a - 4) = 2 \cdot a \cdot (a - 3)}
Duch a kürzen.
[29] 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 2 \cdot (a - 4) + (a - 4) = 2 \cdot (a - 3)}
[30] 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 2 a - 8 + a - 4 = 2 a - 6}
[31] 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 = 8 + 4 - 6 = 6}
Jetzt is' leicht.
[32] 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 = 6}
[33] 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 b = {a \over 2} = 3}
[34] 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 s = a + b = 6 + 3 = 9}
Piketown ist 9 Meilen vom Hotel entfernt.