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(C) The monkey climbs $30$ feet in the first part of the hour and slips $20$ feet,resulting in a net gain of $10$ feet per hour.To reach $120$ feet,we

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$6:00 \, p.m.$

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(C) The monkey climbs $30$ feet in the first part of the hour and slips $20$ feet,resulting in a net gain of $10$ feet per hour.To reach $120$ feet,we must consider the final jump. In the last hour,the monkey will climb $30$ feet and reach the target without slipping back.Therefore,the monkey needs to reach $120 - 30 = 90$ feet first.At a rate of $10$ feet per hour,it takes $9$ hours to reach $90$ feet.Starting at $8:00 \, a.m.$,$9$ hours later is $5:00 \, p.m.$In the next hour (from $5:00 \, p.m.$ to $6:00 \, p.m.$),the monkey climbs $30$ feet,reaching $90 + 30 = 120$ feet.Thus,the monkey touches the flag at $6:00 \, p.m.$

$A$ monkey climbs $30$ feet at the beginning of each hour and rests for a while,during which he slips back $20$ feet before he again starts climbing at the beginning of the next hour. If he begins his ascent at $8:00 \, a.m.$,at what time will he first touch a flag at $120$ feet from the ground?

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