They start riding bicycles towards each other in a plan to meet at the midway point. Each is able to ride at 7 MPH. They live 42 miles apart. One of them has a pet pigeon and it starts flying the instant the boys start traveling. The pigeon flies back and forth at 21MPH between the 2 boys until they meet. How many miles does the pigeon travel?
let's look at the pattern
stage 1
they are 42 miles apart
let the time they meet be after t hrs.
distance travelled by boy 2 = 7t
distance travelled by pigeon = 21t
21t + 7t = 42
28t = 42,
t = 1.5
so the pigeon has travelled 21t or 21(1.5) miles
Stage 2
they are 21 miles apart, ( each boy has gone 7(1.5) miles)
define t the same way as before
distance travelled by boy 1 = 7t
distance travelled by pigeon = 21t
21t+7t=21
t = .75
so the pigeon has travelled 21(1.5) + 21(.75) miles
stage 3
they are 10.5 miles apart (each boy has gone 7(.75) miles, leaving 21 - 2(7)(.75) = 10.5)
distance travelled by boy 2 was 7t
distance travelled by pidgeon was 21t
21t+7t=10.5
t = .375
so the pigeon so far has flown 21(1.5) + 21(.75) + 21(.375)
etc ....
so the total distance covered by the pigeon is
21(1.5) + 21(.75) + 21(.375) + ... to infinity
= 21(1.5 + .75 + .375) + ..
the part inside the bracket is an infinite series with
a = 21.5 , and r = .5
sum = 21 ( 1.5/(1-.5) ) = 21(1.5/.5) = 21(3) = 63 miles
Or, since the boys live 42 miles apart, and they each ride at 7mph, they ride 21 miles each, in 3 hours.
The pigeon, flying at 21 mph for those 3 hours, covers 63 miles.
Steve, Of course!!!!
How very clever of you
aw, shucks. > scuff scuff <
To find out how many miles the pigeon travels, we need to determine how long it takes for the boys to meet.
Let's first calculate the time it takes for each boy to cover the distance. Since they are riding towards each other, we can consider their combined speed, which is 7 MPH + 7 MPH = 14 MPH.
Using the formula: time = distance / speed, we can find the time it takes for each boy to meet.
For one boy, the time taken is: time = 42 miles / 14 MPH = 3 hours
Now, we need to determine how long the pigeon flies. The pigeon's speed is given as 21 MPH.
During the time it takes for the boys to meet (3 hours), the pigeon flies back and forth, covering the total distance traveled by the boys. Since the boys each travel half the distance, the total distance covered by the pigeon is twice that distance.
Total distance covered by the pigeon = 2 * 42 miles = 84 miles
Therefore, the pigeon travels 84 miles.