A cat climbs a 21-meter pole. In the first minute, it climbs 3 meters, and in the second minute, it descends 1 meter. In how many minutes will the cat reach the top of the pole?
Step-by-step Solution:
1. The Key Insight For this classic "slippery pole" problem, the trick is to realize that in the final minute, the cat will climb to the top and won't slide back down. We should first calculate the time it takes to get just short of the final climb.
2. Analyze the 2-Minute Cycle - In the first minute, the cat's progress is +3 meters.
- In the second minute, its progress is -1 meter.
- Therefore, the net progress in one 2-minute cycle is 3 - 1 = 2 meters.
3. Calculate the Time to Get Near the Top - The final climb will be 3 meters. So, we first need to find the time it takes for the cat to reach the (21 - 3) = 18-meter mark.
- Since the cat effectively climbs 2 meters every 2 minutes, the time to climb 18 meters is:
- (18 meters ÷ 2 meters per cycle) × 2 minutes per cycle = 9 cycles × 2 minutes/cycle = 18 minutes.
- At the end of 18 minutes, the cat is exactly at the 18-meter mark.
4. The Final Climb - The very next minute is the 19th minute. This is an odd-numbered minute, which is a climbing minute.
- The cat climbs 3 meters from its position at 18 meters.
- Final position = 18 m + 3 m = 21 m. The cat has reached the top.
Final Answer: The cat will reach the top of the pole in 19 minutes.