Difference between revisions of "NMMRUS 63 Loesung"

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16) <math>{b \over v_k} = t_4 - 30 + 15</math><br/>
 
16) <math>{b \over v_k} = t_4 - 30 + 15</math><br/>
 
17) <math>{b \over v_k} = {b \over v_f} - 15</math><br/>
 
17) <math>{b \over v_k} = {b \over v_f} - 15</math><br/>
−
18) <math>b \cdot ({1 \over v_f} - {1 \over v_k}) = 15</math><br/>
+
18) <math>b \cdot ({1 \over v_f} - {1 \over v_k}) = 15</math><br/><br/>
−
19) <math>b \cdot {{v_k - v_f} \over {v_f \cdot v_k}} = 15</math><br/>
+
19) <math>b \cdot {{v_k - v_f} \over {v_f \cdot v_k}} = 15</math><br/><br/>
 
20) <math>b = 15 \cdot {{v_k - v_f} \over {v_f \cdot v_k}}</math>
 
20) <math>b = 15 \cdot {{v_k - v_f} \over {v_f \cdot v_k}}</math>
  
Line 72: Line 72:
  
 
22)  
 
22)  
−
<math>
+
<math>b =
 
   15
 
   15
 
   \cdot
 
   \cdot
Line 84: Line 84:
 
       1
 
       1
 
     }
 
     }
 +
  }
 +
</math><br/><br/>
 +
 +
23)
 +
<math>b =
 +
  { {
 +
      {15 \cdot (a - 4) \cdot a}
 +
      \over
 +
      {30 \cdot 4}
 +
    }
 +
    \over
 +
    {
 +
      {a - 4}
 +
      \over
 +
      4
 +
    }
 +
  }
 +
</math>
 +
 +
Doppelbrüche auflösen.
 +
 +
24)
 +
<math>b =
 +
  {
 +
    { 15 \cdot (a - 4) \cdot a \cdot 4 }
 +
    \over
 +
    { 30 \cdot 4 \cdot (a - 4) }
 
   }
 
   }
 
</math>
 
</math>
 +
 +
Was geht kürzen.
 +
 +
25)
 +
<math>b =
 +
  {
 +
    { 15 \cdot a }
 +
    \over
 +
    { 30 }
 +
  }
 +
</math><br/><br/>
 +
 +
26)
 +
<math>b =  { a \over 2 }</math>
 +
 +
Bis jetzt wurde es noch nicht erwähnt aber a + b = s . Da setzen wir jetzt 14) und 26) ein.
 +
 +
27) <math>a + {a \over 2} = a \cdot {{a - 3} \over {a - 4}}</math>
 +
 +
Mal 2, mal (a - 4)
 +
 +
28) <math>2 \cdot a \cdot (a - 4) + a \cdot (a - 4) = 2 \cdot a \cdot (a - 3)</math>
 +
 +
Duch a kürzen.
 +
 +
29) <math>2 \cdot (a - 4) + (a - 4) = 2 \cdot (a - 3)</math><br/>
 +
30) <math>2 a - 8 + a - 4 = 2 a - 6</math><br/>
 +
31) <math>a = 8 + 4 - 6 = 6</math>
 +
 +
Jetzt is' leicht.
 +
 +
32) <math>a = 6</math><br/>
 +
33) <math>b = {a \over 2} = 3</math><br/>
 +
34) <math>s = a + b = 6 + 3 = 9</math>
 +
 +
Piketown ist 9 Meilen vom Hotel entfernt.

Revision as of 19:08, 29 October 2007

Wie weit ist es bis Piketown?

zurück zur Aufgabenstellung

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)

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.

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