Question 1207791: Two bees leave two locations 150 meters apart and fly, without stopping, back and forth between these two locations at average speeds of 3 meters per second and 5 meters per second, respectively. How long is it until the bees meet for the first time? How long is it until they meet for the second time?
Found 5 solutions by mananth, AnlytcPhil, ikleyn, greenestamps, Plocharczyk:
Answer by mananth(16946)   (Show Source):
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Bee A speed 3m/s
Bee B speed 5m/s
They are flying towards each other , effective speed = (5+3)=8 m/s
Time taken to meet = d/r = 150/8 =18.75 s (first meet)
In 18.75 s A flew 18.75*3=56.25 m (rate 3m/s)
By then B flew a distance of (5*18.75)=93.75 m
Now A has to fly 93.75m to reach the other end, it will take 93.75/3 = 31.25 s----------(1)
At this point B has flown 31.25*5 = 156.25 (rate 5m/s)
It reached the other end(56.25m) and returned 100m
It means 156.25-56.25 = 100 m from A
The distance between them at this point is 150-100 =50m
They are now facing each other and flying towards each other
effective speed = 8 m/s
t= d/r = 50/8 is the time taken to meet = 6.25 s-----------------------------------(2)
Time taken to meet again after the first meet = 31.25+6.25= 37.50 s
Appears OK
Answer by AnlytcPhil(1807)  (Show Source):
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Their rate of approach is 3+5=8 m/s. So their 150 m distance of separation will
shrink to 0 in 150/8 = 18.75 seconds.
The amount of time from when they are 150 m apart until the next time they will
be 0 m apart
----will always be the same as----
The amount of time from when they are 0 m apart until the next time they will
be 150 m apart
So it will take another 18.75 seconds until they are 150 m apart.
Then it will take another 18.75 seconds for them to get 0 m apart again.
Answer: 18.75 + 18.75 = 37.5 more seconds.
Edwin
Answer by ikleyn(52846)  (Show Source):
You can put this solution on YOUR website! .
Two bees leave two locations 150 meters apart and fly, without stopping, back and forth between
these two locations at average speeds of 3 meters per second and 5 meters per second, respectively.
(a) How long is it until the bees meet for the first time?
(b) How long is it until they meet for the second time?
~~~~~~~~~~~~~~~~~~~~~~
By default, we assume that the bees start at the same time moment.
(a) Question (a) is simple. Initial distance is 150 meters and the rate of approaching is
3 + 5 = 8 m/s. So, the time till the first meeting is 150/8 = 18.75 seconds. ANSWER
At this point, part (a) is complete.
(b) Faster bee covers 150 m in 150/5 = 30 seconds, and after that changes the direction to opposite.
Slover bee covers 150 m in 150/3 = 50 seconds.
So, when the faster bee completes 150 m in 30 seconds, the slower bee is still on the way to its turning point,
and the slower bee will fly additional 50-30 = 20 seconds to get its turning point.
During these 20 seconds, the faster bee, which just changed its direction to opposite,
will cover 20*5 = 100 meters.
So, when the slower bee will reach its turning point, the distance between the bees will be 150 - 100 = 50 m.
Now both bees fly towards each other at the approaching rate of 3+5 = 8 m/s (again).
So, they will cover 50 m in 50/8 = 6.25 seconds.
Thus, the time to meet for the second time is 30 + 20 + 6.25 = 56.25 seconds (counting from the start). ANSWER
It is the same as to say that they will meet in the second time 56.25 - 18.75 = 37.5 seconds after their first meeting moment.
Solved in full, with direct, simple and straightforward consideration, with minimum calculations and with complete explanations.
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
All three responses have the same answer; tutor @ikleyn simply misread the solutions from the other two tutors.
Tutor @ikleyn says it take 18.75 seconds for the two bees to meet for the first time and 56.25 seconds FROM THE START for them to meet the second time.
Both of the other responses say that it take 18.75 seconds for the two bees to meet the first time and THEN ANOTHER 35 seconds for them to meet the second time.
So the three responses are in agreement.
The way I see it, similar to how tutor @Edwin explains it, we can model the solution with a graph showing the two bees "bouncing" between two walls 150m apart. To meet the first time, the two bees simply have to cover the total distance between the two walls, a distance of 150m. But to meet the second time, each bee has to continue to its second wall and then bounce off it and then meet the other bee; to do that the two bees together must travel a distance equal to TWICE the distance between the walls.
So, at a combined speed of 8m/sec, it takes 150/8 = 18.75 seconds for them to meet the first time but then another 300/8 = 37.5 seconds for them to meet the second time.
Answer by Plocharczyk(17)  (Show Source):
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The best way to really understand and explain this problem is to have the bees
flying in a circle of circumference twice 150 or 300, so that each semi-circle
from A to B is 150 m. We are not told the bees must fly in a straight line, so
we are certainly free to assume they fly in this circle. It is 150 circular
meters from A to B. (around the circumference of each semi-circular path.
The slower bee flies clockwise from A, and the faster bee flies
counter-clockwise from B.
In the circles below, the two bees are the dots at the end of the arcs through
which the bees have just traveled. The slower bee is green, and the faster one
red.
In the first circle and third circles below, the bees are meeting. In the
middle circle they are 150 circular meters apart. (You can think of them
as 180o apart, if you like.)
  
Ikleyn above did not understand why I said this:
The amount of time from when they are 150 m apart until the next time they will
be 0 m apart
----will always be the same as----
The amount of time from when they are 0 m apart until the next time they will
be 150 m apart
She wasn't thinking of the bees as if they were flying in a circular or
curved path as I was.
But as you see, the bees travel the same distance in all three circles, so
it will take the same length of time.
In the first circle, their rate of approach is 3+5=8 m/s. So their 150 m
distance of separation will shrink to 0 in 150/8 = 18.75 seconds.
So it will take another 18.75 seconds until they are 150 m apart.
Then it will take another 18.75 seconds for them to get 0 m apart again.
Answer: 18.75 + 18.75 = 37.5 more seconds.
Edwin
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