Mathematics Intersection of Lines
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1 a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement Fund 0-0
2 Intersecton of Lnes
3 Intersecton of Lnes Does the lne shown n the graph pass through the pont (9,5)? A. B. Yes No
4 Soluton Answer: B Justfcaton: The equaton of the lne represents the -value of a pont, gven an -value. If the pont (9,5) s nserted nto the equaton: Snce ths s not true, the pont does not le on the lne.
5 Intersecton of Lnes II The lnes m b and m b ntersect at the pont (, ). Whch one of the followng statements must be true? A. B. C. D. E. m m m m m b b b m Both A andd and or b m m b b
6 Soluton Answer: E (Both A and D) Justfcaton: Snce (, ) s the pont of ntersecton of two lnes, t s a pont that les on both lnes. From the prevous queston, we know that when a pont les on a lne, and can be plugged nto the equaton of the lne. Therefore A. s true. m b and m b Snce s the same n both lnes, the can be equated to gve: D. m b m b
7 Intersecton of Lnes III At what pont do the lnes and ntersect? A. B. C. D. E. 0,,0,, 0,
8 Soluton Answer: B Justfcaton: The -value of the pont of ntersecton can be found b equatng the equatons: Onl B has ths -value. Double check that =0 b nsertng to ether lne equaton. The pont of ntersecton s ,0,0 0 0
9 Intersecton of Lnes IV Whch of the followng lnes ntersects the red lne at the largest -coordnate? A. B. C D. E.
10 Soluton Answer: B Justfcaton: A good wa to solve ths problem s b drawng quck sketches of each lne. Notce that all the lnes ntersect the -as at (0,). B. Startng at the pont (0,), onl a negatve slope wll ntersect the red lne at a postve -coordnate. Ths rules out answers A and D. Compared to C and E, lne B has a much larger run than rse. The graph shows lne B (green) ntersects the red lne at the largest -value. A. D. C. 0 E.
11 Intersecton of Lnes V Jmm s runnng towards Kerr at 7 m/s whle she s runnng awa from hm at 4 m/s. Kerr begns 0 meters awa. How long does t take for Jmm to catch up to Kerr? A. B. C. D. E. s 5 s 0 s 5 s 0 s
12 Soluton Answer: C Justfcaton: Settng up lne equatons s essental to solve ths problem. Let the orgn be the startng pont of Jmm so that we work wth postve numbers. A pont represents the poston of the runners at dfferent tmes. Jmm starts at (0, 0) whle Kerr starts at (0, 0). Snce Jmm moves 7 m ever second, hs poston over tme can be epressed as: Jmm J 7t (where s n meters and t s n seconds) Kerr s poston over tme can be epressed as: snce she stars 0 m awa Kerr 4t 0 When these lnes ntersect, Jmm would have caught up wth Kerr because the wll be at the same poston at the same tme: 7t t t 4t seconds K
13 Intersecton of Lnes VI How far does Jmm travel before he catches up wth Kerr? A. B. C. D. E. 0 m 5 m 40 m 70 m 00m
14 Soluton Answer: D Justfcaton: Remember from the prevous queston that t takes 0 seconds for Jmm to catch up. Snce he moves at 7 m/s, he should travel 70 m. The pont of ntersecton s (0, 70). Notce that Kerr, who runs at 4 m/s, onl travels 40 m n 0 seconds. However, she started 0 meters ahead of Jmm her poston s also at 70 m after 0 seconds. Jmm: 7t 7(0) 70 m Kerr: 4t 0 4(0) 0 70 m
15 Intersecton of Lnes VII The populaton growth of countr X s shown to the rght. Whch of the followng countres wll catch up n populaton the earlest? All the populatons are modelled as lnear relatons. Countr X Year Pop A. B. C. D. Year Pop. Year Pop. Year Pop. Year Pop
16 Soluton Answer: D Justfcaton: Countres A and B wll not catch up n populaton wth Countr X. Ths s because the have a smaller ntal populaton and also ncrease at an equal (Countr A) or slower (Countr B) rate. Countr C ncreases n populaton b 6000 ever 4 ears, whle Countr X ncreases b 4000 ever 4 ears. Countr D has a larger ntal populaton, but ncreases onl b 5000 ever 4 ears. To make calculatons easer, let the varable t represent a change of 4 ears. C: D: t t t t 000t 6000 t 8 000t 9000 t 9 Countr C wll catch up n 06, whle countr D wll catch up n 00. What are the lmtatons of usng a lnear relaton to model populaton growth?
17 Intersecton of Lnes VIII Consder two lnes wth an opposte slope: m b m c At what -coordnate do these two lnes ntersect? A. B. C. D. E. b c b c b c m b c m b c m
18 Soluton Answer: A Justfcaton: Startng at the -ntercept, the two lnes wll approach each other at the same rate snce the have opposte slopes. Intutvel we ma nfer that the two lnes wll ntersect at the mdpont of the -ntercepts. Workng wth the equatons shows that the lnes wll ntersect at the mdpont of b and c. m b m m c b c b m c c b m b m c b b c b
= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]
![= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]](/thumbs/78/76802454.jpg "= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]")
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