An ant climbing up a vertical pole ascends 12 meters and slips down 5 meters in every alternate hour. If the pole is 63 meters high how long will it take it to reach the top?

Options

A. 18 hours
B. 17 hours
C. 16 hours 35 min
D. 16 hours 40 min

Answer

To solve the problem of how long it will take the ant to reach the top of a 63-meter pole while climbing 12 meters and slipping down 5 meters every alternate hour, we can break it down step by step.

### Step-by-Step Solution:

1. **Understand the Climbing and Slipping Pattern:**
- The ant climbs 12 meters in the first hour.
- In the second hour, it slips down 5 meters.
- Therefore, in every 2-hour cycle, the net gain in height is:
\[ \text{Net gain in 2 hours} = 12 \text{ meters (up)} - 5 \text{ meters (down)} = 7 \text{ meters} \]

2. **Calculate the Number of 2-Hour Cycles Needed:**
- The total height of the pole is 63 meters.
- After each 2-hour cycle, the ant climbs 7 meters.
- To find out how many complete 2-hour cycles it takes to get close to the top, we can calculate:
\[ \text{Number of cycles} = \frac{63 \text{ meters}}{7 \text{ meters/cycle}} = 9 \text{ cycles} \]
- This means it will take 18 hours to reach 63 meters if it were to continue this pattern.

3. **Determine the Position After 9 Cycles:**
- After 9 cycles (18 hours), the ant will have climbed:
\[ 9 \text{ cycles} \times 7 \text{ meters/cycle} = 63 \text{ meters} \]
- However, we need to check if it reaches the top during the 9th cycle.

4. **Climbing in the 19th Hour:**
- In the 19th hour, the ant climbs an additional 12 meters:
\[ \text{Height after 19 hours} = 63 \text{ meters} + 12 \text{ meters} = 75 \text{ meters} \]
- Since the pole is only 63 meters high, the ant will reach the top before slipping down.

5. **Calculate the Exact Time to Reach the Top:**
- After 18 hours, the ant is at 56 meters (after slipping down).
- In the 19th hour, it climbs to 68 meters, which exceeds the height of the pole.
- The remaining distance to climb after reaching 56 meters is:
\[ 63 \text{ meters} - 56 \text{ meters} = 7 \text{ meters} \]
- The time taken to climb the remaining 7 meters at a rate of 12 meters per hour is:
\[ \text{Time} = \frac{7 \text{ meters}}{12 \text{ meters/hour}} = \frac{7}{12} \text{ hours} = 35 \text{ minutes} \]

6. **Total Time Taken:**
- Therefore, the total time taken by the ant to reach the top of the pole is:
\[ 18 \text{ hours} + 35 \text{ minutes} = 18 \text{ hours} 35 \text{ minutes} \]

### Final Answer:
The ant will take **18 hours and 35 minutes** to reach the top of the 63-meter pole.