Mathematik für Biologen (Ri), Ergänzungen 1. Lösungs-Skizzen.

(1) Vogelflug. Hier ein Schnitt durch die Erdkugel, und zwar geht der rechte Halbkreis gerade durch alle Orte mit östlicher Länge 20^{∘}.

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Der Winkel Belgrad - Erdmittelpunkt - Kapstadt ist α + β = 80^{∘}, das Bogenmaß von 80^{∘} ist a = \frac{2π}{360^{∘}} · 80^{∘} = 1, 396. Demnach ist die Bogenlänge a · r = 1, 396 · 6 370 km = 8 894 km. Der Vogel fliegt also rund 9 000 km.

(2) Erdumdrehung. In 24 Stunden dreht sich die Erde einmal um ihre Achse, also um 360^{∘}. In 1 h dreht sie sich um (\frac{360}{24})^{∘} = 15^{∘}. Das Bogenmaß von 15^{∘} ist \frac{2π}{360} · 15 = 0, 262. Also ist die Länge des entsprechenden Kreisbogens 0, 262 · 6 370 km = 1 667 km. Die Geschwindigkeit eines Punkts am Äquator ist also 1 667 km/h — ganz schön schnell!

(3) Uhr. Wir beginnen um Mitternacht und fragen, welchen Winkel der Minutenzeiger nach t Minuten zurückgelegt hat.

60 Min ≃ 360^{∘} = 2π
1 Min ≃  6^{∘} = \frac{1}{60} · 2π = \frac{π}{30}
t Min ≃ t· 6^{∘} = t · \frac{π}{30}

Entsprechend der Stundenzeiger:

12 h ≃ 360^{∘} = 2π
1 h ≃  30^{∘} = \frac{1}{12} · 2π = \frac{π}{6}
1 Min ≃  (\frac{1}{2})^{∘} = \frac{π}{60·6} = \frac{π}{360}
t Min ≃ t · (\frac{1}{2})^{∘} = t · \frac{π}{360}

Frage: Wann stehen nach Mitternacht Minuten- und Stundenzeiger zum ersten Mal senkrecht aufeinander? Antwort: Nach t Minuten! Wir müssen nur noch t ausrechnen. Der Minutenzeiger hat nach t Minuten den Winkel t\frac{π}{30} zuückgelegt, der Stundenzeiger den Winkel t\frac{π}{360}. Der Winkel zwischen beiden ist \frac{π}{2} , also

\frac{π}{2} = t\frac{π}{30} − t\frac{π}{360} = tπ(\frac{1}{30} − \frac{1}{360}) = tπ \frac{11}{360}.

Also t = \frac{360}{22} = 16, 4. Nach 16, 4 Minuten (also 16 Minuten und 24 Sekunden) stehen die beiden Zeiger senkrecht aufeinander (und dies zum ersten Mal nach Mitternacht).

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