Illustrierende Aufgaben zum LehrplanPLUS. Working at the zoo

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1 Working at the zoo Stand: Jahrgangsstufen FOS 11, BOS 1 Fach/Fächer Übergreifende Bildungs- und Erziehungsziele Zeitrahmen Benötigtes Material Mathematik Sprachliche Bildung ca. 30 Minuten zugelassener Taschenrechner, ggf. Wörterbuch Kompetenzerwartung Lehrplan Mathematik FOS 11 LB Lehrplan Mathematik BOS 1 LB Die Schülerinnen und Schüler ermitteln die größtmöglichen Intervalle, in denen der Graph einer ganzrationalen Funktion jeweils gleiches Monotonieverhalten bzw. Krümmungsverhalten aufweist. Dafür berechnen sie Ableitungen, insbesondere mit der ihnen bekannten Ableitungsregel. Weiterhin begründen sie damit die Existenz von relativen Extrempunkten und Wendepunkten. Sie bestimmen ferner Art und Koordinaten solcher Punkte. Seite 1 von 5

2 Aufgabe To earn some extra money you work at the zoo during weekends and in your holidays. Two months ago an ice bear baby was born in the zoo. This event and the good weather have led to an unprecedented increase in the number of visitors, which in turn has presented a challenge for the zoo personnel. In order to optimize personnel planning, the visitor figures have been monitored over the course of several weekends. As you have asked the zoo manager for a more challenging job, he has offered you to support the group responsible for personnel planning. So far they have already come up with a function, given by the function term below, to model the approximate development of the number of visitors present at any moment during the opening hours (from 8 a.m. to 7:30 p.m.) on weekends. ( ) =- ( ) Π[ ] f x 5 x 3.5x 93.5x 780 ;x 8;19.5 ;xinhours after midnight For the reasons mentioned below, the zoo manager wants you to provide answers to the following questions: 1. What is the maximum number of visitors present in the zoo? This information is necessary to make sure that the security requirements for zoos can be fulfilled at all times during the opening hours.. At what time is the highest rate of change of the number of visitors present in the zoo? At that time all the box offices at the entrance should be opened. o How many people per hour want to enter the zoo at that time? o Would it make sense to build a second entrance to reduce waiting times, considering that the zoo currently has eight box offices? 3. During what period of time do all shops, kiosks and restaurants need to be open if this is when more than 4500 people are in the zoo? Additional question: 4. How many seats would actually be necessary for the shows in the dolphinarium, which take place at 11:00 a.m., :00 p.m. and 3:30 p.m. if it can be assumed that about one quarter of the people present in the zoo visit the show? What conclusions can be drawn from your results? Seite von 5

3 Hinweise zum Unterricht Die Lösungsvorschläge unter den Hinweisen zum Unterricht erfolgen im Folgenden stichpunktartig. Diese sind nicht als vollständige, alternativlose Lösungserwartung zu sehen. Auch von einer strengen mathematischen Fachnotation wird hier abgesehen. Arbeitsform: Gruppenarbeit (max. 3 Schüler), Partnerarbeit oder Einzelarbeit möglich. Anmerkung: Der englischen Konvention folgend werden in dieser Aufgabe zur Darstellung von Dezimalbrüchen Punkte statt Kommas verwendet. possible solution: 1. ( ) =- ( - + ) f x =-5 x - 3.5x x- 780 ;D = [8;19.5] f'x 5 3x 65x ( ) f'x = 0 3x -65x- 93.5= 0 65± ± 703 x1/ = = 3 6 x ª 15.5; x ª 6.41œD f ( ) =- ( - ) ( ) ª f'' x 5 6x 65 f'' 15.5 = =- 66.5< 0 f f The number of visitors present in the zoo reaches its maximum at about 7900 people.. f'' ( x) =-5( 6x-65) f'' ( x) = 0 ( 6x- 65) = 0 65 x= ª f''' x =-5 6= ( ) At about 10:50 a.m. all box offices should be opened, because at that time the rate of change of the number of visitors present in the zoo is the highest. Ê65ˆ Ê Ê65ˆ 65 ˆ f' Á =- Á - + ª 6 Ë Á Ë 6 Ë Seite 3 von 5

4 At that time about 1470 people per hour want to enter the zoo. 3 Assuming that it takes about 45 seconds per visitor to buy a ticket, 60: = 80 people can buy 4 a ticket per hour per box office. That means that 8 80= 640 people can buy a ticket per hour ª In order to avoid waiting times entirely about 18 box offices would be necessary. Considering the fact that so many box offices would only be needed during peak hour and that the highest number of visitors per hour who want to enter the zoo is surely significantly lower during the week and the winter months, it is questionable if it is worth building a second entrance. 3. The following inequation needs to be solved: f( x) > 4500 graphical solution: From about 11:30 a.m to 6:10 p.m. more than 4500 visitors are in the zoo. figure 1: number of visitors present in the zoo 4. f11 = ª ª 956 For the presentation at 11 a.m. about 956 seats would be necessary. Seite 4 von 5

5 f14 = ª ª 1856 f15.5 = ª ª 1969 For the presentation at p.m. about 1860 seats would be necessary and for the one at 3:30 p.m. even about 000. It is rather questionable that there is room for so many seats in the dolphinarium. Therefore it would be a good idea to leave some standing room or to offer more shows. Anregung zum weiteren Lernen Der Term der Funktion, die die Besucherzahl im Park in Abhängigkeit von der Tageszeit beschreibt, könnte von den Schülern bei Vorgabe geeigneter Werte auch selbst bestimmt werden. Weitere Fragestellungen im Kontext der Aufgabe sind denkbar. Seite 5 von 5