How Many Jumps Does a Monkey Need to Climb an Oily Pole?
Imagine a curious little monkey trying to climb an oily 30-meter pole with an interesting twist: in every jump, it climbs 4 meters but then slides back 2 meters. This scenario raises a fascinating question: how many jumps does the monkey need to reach the top of the pole? In this article, we'll break down the problem and solve it step-by-step using clear examples and detailed explanations.
Understanding the Progress per Jump
Let's start by analyzing the monkey's progress with each jump:
Each time the monkey makes a jump, it climbs 4 meters but then slides back 2 meters. Consequently, the net progress per jump is:
Net progress 4m - 2m 2m
Calculating Effective Height Before the Last Jump
Now, we need to determine how high the monkey can go without reaching the top so that it can successfully make the final jump. Let's do the math:
If the monkey climbs 2 meters with each jump, it will reach a height of 28 meters after 14 jumps (since 14 jumps x 2 meters/jump 28 meters).
However, on the 15th jump, the monkey climbs an additional 4 meters and doesn't slide back. So, the monkey would reach 32 meters in total, surpassing the 30-meter mark and successfully climbing the pole.
A Closer Look at the Monkey's Jumps
Let's break it down even further:
After 6 jumps, the monkey reaches 18 meters (since 6 jumps x 2 meters/jump 12 meters 4 meters on the 6th jump). After 7 jumps, the monkey reaches 21 meters (since 7 jumps x 2 meters/jump 4 meters on the 7th jump 14 meters 4 meters). After 8 jumps, the monkey reaches 25 meters (since 8 jumps x 2 meters/jump 4 meters on the 8th jump 16 meters 4 meters). On the 9th jump, the monkey climbs the remaining 5 meters and reaches the top without sliding back.So, by the 9th jump, the monkey climbs the final 5 meters and makes it to the top of the pole.
Another Perspective on the Problem
Another way to tackle this problem is to consider the net height climbed in each jump:
Initially, the net length of the monkey's jump can be seen as 5 meters (since it climbs 4 meters and slides back 2 meters, the net progress is 5 meters - 2 meters 3 meters).
Using this approach, the height of each jump can be calculated as follows:
Height of 1st jump 5 meters (since it hasn't slid back yet). Height of 2nd jump 8 meters (5 meters 3 meters). Height of 3rd jump 11 meters (8 meters 3 meters). Height of 4th jump 14 meters (11 meters 3 meters). Height of 5th jump 17 meters (14 meters 3 meters). Height of 6th jump 20 meters (17 meters 3 meters). Height of 7th jump 23 meters (20 meters 3 meters). Height of 8th jump 26 meters (23 meters 3 meters).At this point, the monkey is only 4 meters below the top of the pole, so it just makes the final jump to reach the top.
Conclusion
The monkey needs a total of 9 jumps to reach the top of the 30-meter pole. Each jump involves climbing 4 meters and then sliding back 2 meters, except for the final jump, which is a direct climb to the top.