Jack & Jill
Math brain teasers require computations to solve.
Jack and Jill are climbing down a vertical wall. They both use ropes hanging from the ceiling of a huge room. They start at the same time from the top of the wall which is 16 meters tall. The race is divided into two stages; the first 8 meters and the last 8 meters. A penalty of half a minute waiting time is placed upon whoever loses the first stage (first 8 meters) and it commences as soon as the winner finishes the first 8 meters.
The following describes their climbing techniques:
1. Jack climbs down in a zigzag way (moving one step downwards diagonally at a fixed angle from the horizon line). In each step movement he completes the distance of 1 meter diagonally. Initially, his feet are distant 60 cm apart aligned on a perfectly horizontal line. It takes him 15 seconds to complete each move. Jack has to rest for 1 minute for every 8 meters he vertically completes.
2. Jill hops down according to a perfect half circular arc movement. In each hop, she moves a distance of 1.57 meters along the arc path (not the vertical). It takes her 15 seconds to complete each hop. Jill has to rest for 1 minute for every 6 meters she vertically completes.
Who will reach the ground first and how many additional minutes will it take the loser to finish the race?
Answer
Jack will win the race and it will take an additional half minute to catch up with him.
Jacks moves 0.8 meters vertically in each step movement:
According to Pythagoras theorem for right angled triangle:
A^2 = B^2+C^2 (where A is the hypotenuse)
Replacing A by the diagonally crossed distance, B by the distance between Jack's feet and C by the vertically climbed distance:
1^2 = 0.6^2+C^2
C = 0.8 m
Jill moves 1 meter vertically in each hop.
According to the circle circumference calculation:
Circumference = Diameter * Pi
(where Pi = 3.14 and Diameter is the vertically climbed down distance)
2 * 1.57 = D * 3.14
Diameter = 1 meter
1.Time: 1.5 minutes elapsing
Jack: Finished 4.8 meters
Jill: Finished 6.0 meters and starting her 1 minute rest
2.Time: 2.5 minutes elapsing
Jack: Finished 8.0 meters (crossed an additional 3.2 meters in 1 minute while Jill is resting)
Jill: Finished 6.0 meters and ending her 1 minute rest
Note: Losing the first stage (8 meters), Jill has to wait for half minute penalty time and at the same time Jack starts his 1 minute rest having crossed 8.0 meters.
3. Time: 3.0 minutes elapsing
Jack: Finished 8.0 meters and still resting for another half a minute
Jill: Finished 6.0 meters and resuming her climb down having finished the half
minute penalty
4.Time: 3.5 minutes elapsing
Jack: Finished 8.0 meters and ending his 1 minute rest
Jill: Finished 8.0 meters (crossed an additional 2.0 meters in half minute while Jack is resting)
5. Time: 4.5 minutes elapsing
Jack: Finished 11.2 meters
Jill: Finished 12.0 meters and starting her 1 minute rest
6.Time: 5.5 minutes elapsing
Jack: Finished 14.4 meters (crossed an additional 3.2 meters in 1 minute while Jill is resting)
Jill: Finished 12.0 meters and ending her 1 minute rest
7.Time: 6.0 minutes elapsing
Jack: Finishing the race (16 meters) as the winner
Jill: Finished 14.0 meters and needing another two meters to finish the race which will take her an additional half minute
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