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 $8:00\text{a.m.}$, at what time will he first reach a flag positioned $120$ feet above the ground?

Let's analyze the net distance covered by the mountaineer in each hour cycle:
* Distance ascended in the first part of the hour = $30$ feet
* Distance slipped back during rest = $20$ feet
* Net effective progress per hour = $30-20=10$ feet

The total distance to reach the flag is $120$ feet.
Crucially, in the final hour of climbing, once he reaches the top ($120$ feet) during his $30$ -foot ascent phase, he will touch the flag before he ever slides back.

Remaining distance before this final $30$ -foot dash:

$$120-30=90\text{ feet}$$

Time taken to cover the first $90$ feet at the net rate of $10\text{ feet/hour}$:

$$\text{Time}=\frac{90}{10}=9\text{ hours}$$

Since he starts at $8:00\text{a.m.}$, after exactly $9$ hours, the time will be:

$$8:00\text{a.m.}+9\text{ hours}=5:00\text{p.m.}$$

At $5:00\text{p.m.}$, he has completed $9$ complete cycles and is at a height of $90$ feet.

At the start of the $10\text{th}$ hour (from $5:00\text{p.m.}$ onwards), he begins his next ascent phase of $30$ feet:

$$\text{New Height}=90+30=120\text{ feet}$$

He completes this ascent at the peak of the $10\text{th}$ hour interval cycle. Since each full cycle lasts $1$ hour, he completes the ascent and reaches the flag at $6:00\text{p.m.}$