(a) How far, in metres, did Ann travel in the first 2 minutes?

Question

Two villages, P and Q, are joined by a straight road 6000m long. Ann left P and ran to Q at a steady speed of 3m/s. At the same instant that Ann left P, Ben left Q cycled to P at a steady speed of 7m/s.

(a) How far,, in metres, did Ann travel in the first 2 minutes?

(b) Calculate the distance between Ann and Ben at the end of the first 2 minutes.

(c) Ann and Ben passed each other at M.
Calculate the Distance PM.

(d) Calculate the time that Ben took to cycle from Q to P. Give your answer in minutes and seconds, correct to the nearest second.

(e) The villages appear on a map which has a scale of 2cm to 5km.
(i) Express this scale in the form 1:n.
(ii) Calculate the length of the joining P and Q on the map.

Answer 1

All these can be answered using the basic formula for distance of:
Distance = rate x time
D = rt

a) Ann's rate is 3 m/s and her time is 2 minutes, but we need that in seconds.
r = 3 m/s
t = 2 min x 60 sec/min = 120 sec

Then it is easy to calculate the distance as:
D = 3 x 120
= Ann will have travelled 360 metres after 2 minutes

b) Ann and Ben are closing at a rate of 10 m/s (3+7)
r = 10 m/s
t = 120 seconds (2 minutes)
D = 10 x 120 = 1200
So they have travelled 1200 meters (360m for Ann, 840m for Ben).
Subtracting 6000 - 1200, you get 4800m.
So they are 4800 meters apart after 2 minutes.

c) Again, they cover 10 m/s together, so
t = D/r
t = 6000 m / 10 m/s
t = 600 seconds
t = 10 minutes
In 10 minutes, Ann would travel:
Distance from Point P to M = 3 m/s x 600 sec
D = 1800 metres from P to M

d) Distance = 6000, rate = 7 m/s
t = 6000 / 7
t = 857 1/7 seconds

Now convert this to minutes and seconds:
857 / 60 = 14 r. 17
So it took Ben 14 minutes 17 seconds (actually 17.142857... seconds), but they said to the nearest second.

e) To determine the ratio of scales, you need to get them to the same units.
2 cm : 5 km
1 cm : 2.5 km
1 cm : 2500 m
1 cm : 250000 cm
(i) 1:250000 scale

From above we know that 1 cm is equivalent to 2.5 km. Write these as similar ratios.
1 / 2.5 km = x / 6 km
x = 6 / 2.5
x = 2.4 cm
(ii) 6km would be represented as 2.4 cm on the map.

In summary:
(a) 360 m
(b) 4800 m
(c) 1800 m
(d) 14 minutes 17 seconds
(e:i) 1:250000
(e:ii) 2.4 cm

Answer 2

(a) V ann = constant = 3 m/s
S ann in 2 min = V * T = 3 * 2 * 60 = 360 m.

(b) S ann in 2 min = 360 m. -> 360 m from P
S ben in 2 min = V * T = 7*2*60 = 840 m. -> 840 from Q
S ann - S ben = 6000-840-360 = 4800 m.

(c) as Ann and Ben meets each other at M, then the distance of Ann and Ben is equal to each other with relative to any of the 2 points P or Q , then :
3*T = 6000 - 7*T
10T = 6000
T = 600 sec. then they meet each other after 600 sec.
PM (from P) = 3*600 = 1800 m. = 6000 - 7*600
PM (form Q) = 7*600 = 4200 m. = 6000 - 3*600

(d)T=S/V = 6000/7 = 857 sec. = 857/60 min. = 14.28 min.

(e)(i) 1 : 2.5 ( divide the scale by 2 to get 1 on the L.H.S ).
(e)(ii) S(actual) = 6000 m = 6 Km.
S(map scale) = 6/2.5 = 2.4 cm.

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Answer 3

(a) 3 m/s * 60s/min * 2 min = 360 m
(b) Ben traveled 7*60*2 = 840 m; distance between the two was 6000 - (360 + 840) = 4800 m
(c) In x minutes Ann traveled 180x meters and Ben traveled 420x meters; they passed each other at the minute that 180x + 420x = 6000; x = 10 min; distance PM is distance Ann traveled in 10 minutes = 180(10) = 1800 m
(d) Ben traveled 420x meters per minute; to travel 6000 m took 6000 / 420 = 14.28 min = 14 1/7 min = 14 min 17 sec
(e) (i) 2 cm:5 km = .02 m:5000 m; multiply each term by 50 to get 1:250000
(e) (ii) If the scale is 1:250000 and PQ = 6000 m then PQ on the map will be 6000 / 250000 = .024 m = 2.4 cm

Answer 4

(a) (3 m/sec)(2 min)(60 sec/min) = 360 meters
(b) Ben's distance= 7(2)(60) = 840 meters
Distance between Ben and Anna = 6000 - (360+840) = 4800 meters
(c) The time for anna and ben to meet is the same, while PM + MQ = PQ = 6000 meters
At that time t, Anna has travelled 3t meters, while ben has travelled 7t meters.
3t + 7t = 6000
10t=6000
t=600 seconds or 10 minutes
Anna travelled PM = 1800 meters
(d) Ben's time = 6000 m divided by 7 m/s = 857.14 s or 857 sec or 14 min 17 sec
(e)(i) 1cm:2.5km
(ii) 2.4 cm

Answer 5

A. 3m/s x 60s/m x 2m

B 6000 - (ans A + (7m/s x 120s)

C PM = 6000 x (2 / 7)

D (6000m / 7m/s ) / 60

E i 5/2 x 100 x 1000
ii 6000/5 = (Ei)/2

Answer 6

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