A mountaineer ascends 30 feet at the start of each hour, then pauses to rest and slides back 20 feet before beginning his climb again at the start of the next hour. If he starts his climb at 8:00 a.m., at what time will he first reach a flag positioned 120 feet above the ground ?

Given:

The mountaineer starts his climb at = 8:00 a.m.

The mountaineer ascends at the start = 30 feet each hour

The mountaineer pauses to rest and slides back = 20 feet each hour

Therefore, Mountaineer net gain = (30 - 20) ⇒ 10 feet each hour

So,

Time taken to cover 90 feet = 90 ÷ 10 ⇒ 9 hours 

Time taken to cover the remaining (120 - 90) 30 feet = 1 hour   

Total Time taken to cover 120 feet = 9 hours + 1 hour ⇒ 10 hours

Therefore, 

The time the he will first reach a flag positioned 120 feet above the ground = 8:00 a.m. + 10 hours ⇒ 6:00 p.m.

Hence, "Option 3" is the correct answer.

Additional Information1) As it is given that the mountaineer ascends 30 feet at the start of each hour, the last distance of 30 feet to reach 120 feet, there will be no sliding back to climb again. And thus the first distance only i.e. 120 feet - 30 feet = 90 feet is calculated considering the net gain.

2) The rest i.e. 30 feet covers in 1 hour as already given.