Let's clarify the climbing scenario involving Rumi and Noam:
- Rumi starts at the bottom and climbs 10 feet per minute.
- Noam starts 10 feet up and climbs 8 feet per minute.
Now, let's analyze how their positions change over time.
Rumi's Position
Rumi starts at the bottom (0 ft) and climbs at a rate of 10 feet per minute. After \( t \) minutes, Rumi's height can be expressed as: \[ R(t) = 10t \text{ feet} \]
Noam's Position
Noam starts 10 feet up (10 ft) and climbs at a rate of 8 feet per minute. After \( t \) minutes, Noam's height can be expressed as: \[ N(t) = 10 + 8t \text{ feet} \]
Finding when Rumi catches up to Noam
To find out when Rumi catches up to Noam, we set their height equations equal to each other: \[ 10t = 10 + 8t \]
Now, let's solve for \( t \):
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Subtract \( 8t \) from both sides: \[ 10t - 8t = 10 \] \[ 2t = 10 \]
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Divide both sides by 2: \[ t = 5 \]
Conclusion
After 5 minutes, Rumi will catch up to Noam. We can calculate their height at that time:
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Rumi's height: \[ R(5) = 10 \times 5 = 50 \text{ feet} \]
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Noam's height: \[ N(5) = 10 + 8 \times 5 = 10 + 40 = 50 \text{ feet} \]
Thus, both Rumi and Noam will be at 50 feet after 5 minutes.