## Wednesday, June 8, 2011

### June Vacation Chap 9.2 RATE (L5) Can he catch up with her?

Grace left home at 6.30 am to walk to school. 10 minutes later, her brother Shamus saw her wallet on the dining table. He grabbed the wallet and hopped onto his bicycle and rode after Grace.

Shamus rode his bicycle 3 times as fast as Grace walked.
Q1: What time was it when Shamus caught up with Grace?
Q2: If the school is 800m from home, did Shamus reach Grace before she arrived at school?

Use the following to guide you...
• What information do you know from the problem?
• What else do you need to know to solve the problem?
• Pick a reasonable number for the information you need.

1. Q1 6.45 or it can be Achilles and the Tortoise paradox ,because the pursuer must always reach the point where the pursued started so Grace will always ahead of Shamus.

Q2 It depends on the speed that Grace walks at.
Shamus can catch up with her if her velocity is 60m/min.
OR
Shamus will not be able to catch up with her if her velocity s 100m/min

2. Q1)

Grace would had walked 10 minutes before Shamus started. Shamus would take 3 and a third minute Grace walked in 10 minutes.

1 Minute After Realising :

Grace would had walked 11 minutes while Shamus had covered 3 minutes of Grace's distance.
Difference in distance (measured in minutes) : 11 minutes - 3 minutes = 8 minutes.

2 Minutes After Realising :

Grace would had walked 12 minutes while Shamus had covered 6 minutes of Grace's distance.
Difference in distance (measured in minutes) : 12 minutes - 6 minutes = 6 minutes.

3 Minutes After Realising :

Grace would had walked 13 minutes while Shamus had covered 9 minutes of Grace's distance.
Difference in distance (measured in minutes) : 13 minutes - 9 minutes = 4 minutes.

Pattern : For every minute Shamus cycle, he gets closer to her sister, Grace, every 2 minutes.

Difference when first started : 10 minutes
10 minutes / 2 minutes = 5 mins.

5 minutes after 6.40AM would be 6.45AM

Q2) Grace must be walking slower or at 53 and a third per minute in order for Shamus to catch up with her.

3. Answering this question with simple listing.

=============================

Shamus speed is 1 X 3 = 3
as his speed is three times faster.
============================

based on grace timing FIrst minute and so on.
Shamus : 3 6 9 12 15 Grace : 11 12 13 14 15
=======================

6.30pm ----->10mins and 5mins

it depends on grace walking speed, if grace walks at a speed of 100m/min, there is no way Shamus could actually catch up.

4. Q1) Shamus' speed is 3 times that of grace, so what grace covers in 3 minutes Shamus covers in 1 minute. Since Grace has had a 10 minute head start we shall count from there.

Grace: 10, 11, 12, 13, 14, 15

Shamus: 0, 3, 6, 9, 12, 15

According to the data above this means that it will take 15 minutes for Shamus to catch up with Grace.

The answer will be: 6:45 am

Q2) 800m / 15 minutes = 53.3 (rounded off) meter per minute

If Grace walks faster than that speed, Shamus wouldn't be able to catch her

If Grace walks slower or equal to that speed, Shamus would be able to catch her before she reaches school

5. 1) Shamus' speed is 3 times as fast as Grace. So we can list the walking speed of Grace and Shamus.
Grace:10,11,12,13,14,15
Shamus:0,3,6,9,12,15
So Shamus would catch up with Grace 15 minutes after she left home(6.45 am).

2) It depends on the walking speed of Grace.

6. 1) Assume: Grace speed is 10m/min, so if Shamus is 3 times faster then Grace's walking speed, his speed would be 30m/min. So Grace would have already walked 100m in her first 10 minutes. Shamus' speed is 20m/min faster then Grace's, that difference must be travel 100m in order to reach Grace, thus : 100m / 20m = 5 min. Finally 5 min + 10 min ( the time when shamus realize that Grace left her wallet) =15 min. 15min after 6.30 am would be 6.45 am.

2) It depends of Grace's speed.

7. Q1) 1 clue: Shamus is travelling 3 times faster than Grace.
2 clue: Time of Shamus will reach Grace after 6 40 and above.
( Time Grace started walking + 10 minutes later ) 6 30 ___10 mins___6 40

Grace : Shamus
Time started: 6 40am : 6 30am
(1times x time taken(y) ) : (3 times x time taken(y) )
Time ended: same(?)(x) : same(?)(x)

from the data, we can say that we need to find the "time ended". They started with 10 minutes different.The speed ratio is 1 : 3. Lets take "time ended" as 'x'.
Grace: (x) = (6 40am) + (1 x time taken(y))
Shamus: (x) = (6 30am) + (3 x time taken(y))

(6 30am) + (3 x (y)) = (6 40am) + (1 x (y))
(3 x (y)) = (10mins) + (1 x (y))
(2 x (y)) = (10mins)
(y) = (5 mins)
(time taken) = (5mins)
Grace: (6 40am) + (1 x time taken(y))
(6 40am) + (1 x 5mins)
= 6 45am
Time they meet 6 45am

Q2) Even though it only takes 5 mins for Shamus to reach Grace, but it really matter how fast Grace travel if by that time Grace is very near the school Shamus would not be able to catch up with her. But is Grace is travelling slower, Shamus would be able to catch up with Grace.

8. Information 1)10 minutes later .2)Shamus rode his bicycle 3 times as fast as Grace walked.
I still need the speed of Grace walking .

__________________________________________________

Shamus is 3 times faster than grace .
Every 1 min of the distance that Shamus travel ,

Q1) Grace needed 3 min.from that , it have a ratio of Shamus : Grace is 3 : 1
Grace have a started 10 mins early so the ratio became 3 : 11
Shamus)0 3 6 9 12 15 }distance
Grace )10 11 12 13 14 15 }distance
TOTAL: it took 5 mins to catch up
__________________________________________________

Q2) (a) If Grace speeds is 53.3 m/min and below , Shamus will able to catch up

(b) If Grace speeds is 54.0 m/min and above , Shamus will not be able to catch up

9. 1) Shamus speed is thrice the speed of Grace, thus the distance covered by Grace per 3 min is what Shamus covers in 1min.

Grace has a 10min head start.
Shamus: 0, 3, 6, 9, 12, [15] (mins)
Grace : 10, 11, 12, 13, 14, [15] (mins)

6hr 30min + 15min = 6hr 45min = 6.45am
The time for Shamus to meet up with Grace is 6.45pm.
________________________________________________________________

2) This actually depends on the speed of Grace.
If Grace walks at 57m/min, then Shamus will not be able to catch up.
However, if Grace walks slower than 56m/min, Shamus will be able to catch up.

10. This comment has been removed by the author.

11. 1)Shamus speed:grace speed=3:1
=the distance covered by grace in 3min is what shamus covers in 1min.
grace has a 10 min headstart.

so use a simple listing method:
shamus:0,3,6,9,12,(15)
Grace:10,11,12,13,14,(15)
10min+5min=15min.

6.30am 10min headstart 6.40am 5min to catch up 6.45am
|-------------------------------|-------------------------|

shamus will meet grace at 6.45am

2)depends on speed of grace walking.if she walks at the average speed of people walking,which is 1.2m/s,shamus would not be able to catch up.if she walks at 0.9m/s,shamus would be able to catch up.

12. grace walked 10 min+5 min=10a+5a=15a
shamus cycle 3x fasterand started later than grace by 10 min so is =3x5a=15a
so the answer is=6.30am + 15 min=6.45am
B)answer is unknown due to unknown speed

13. Assume Grace's speed is x m/s
then, Shamus' will be 3x m/s
Grace left home at 0630, Shamus found her wallet after 10 minutes.
Let x be 1.5 m/s
hence, Grace's speed is 1.5 m/s while Shamus' speed is 4.5 m/s.
Hence, Grace has a 900m head start.
For every 1 sec, Shamus catches up by 3m.
Shamus will catch up Grace after 15 minutes
Ans. 0645
--------------------------------------------------
No.
Assuming that Grace's speed is 1.5m/s Shamus ill not catch up to Grace in time.
--------------------------------------------------

14. grace: d = s x t

Shamus: d = 3s(t -10)
st = 3s(t-10)
t = 3t - 30
2t = 30
t = 15 mins

time caught up = 6.45am

school is 800m from home.
grace will take 800/5 min to reach school
if shamus were to reach grace before she arrived tat school, then 800/5 > 15

800/15 m/min = 800/(15 x 16) m/sec
= 40/45 m/sec

Shamus will not reach her before she arrived at school

15. 1. 6.45pm or based on the paradox by Archilles, Shamus will not ever catch up as Grace is always a third ahead of Shamus.

2. It depends on the walking speed of Grace and the variables that might take place while she is walking. For example a traffic light stop or the energy Grace has.

16. 1) Since Grace has a 10min head start, we would start counting from multiples of 10, but for shamus since he travels at 3 times her speed we would count from multiples of 3. their LCM would be 15. therefore he would catch up with her at 6:45am.

2)It depends as we do not know Grace's walking speed, and there are other factors that might affect her walking speed (e.g. traffic light, accident, etc)

17. let X be grace speed and 3X be shamus speed
1)
6.40pm Grace:10X Shamus:0X
6.41pm Grace:11X Shamus:3X
6.42pm Grace:12X Shamus:6X
6.43pm Grace:13X Shamus:9X
6.44pm Grace:14X Shamus:12X
6.45pm Grace:15X Shamus:15X
shamus will catch up with grace at 6.45pm

2)yes

18. 1) Grace's speed = a/min
Shamus speed = a/min x 3 =3a/min

Since 10 minute has passed, Grace has travelled 10a distance.

Grace:Shamus
10a 0a ----6.40pm
11a 3a ----6.41pm
12a 6a ----6.42pm
13a 9a ----6.43pm
14a 12a ---6.44pm
15a 15a ---6.45pm

Ans: 6.45 pm

2) It depends on what is Grace's speed.

19. 1) The speed of Grace is x/min while Shamus' speed is 3x/min. As Grace travelled for 10 min, she would have travelled 10x.
10x, 0x
11x, 3x
12x, 6x
13x, 9x
14x, 12x
15x, 15x

This depends on both their speeds.