To find the range of the function that represents the total amount of paper and cardboard recycled at the plant over 8 hours, we can first break down the problem into two parts: the amount recycled in the first hour and the amount recycled during the next 7 hours.
- First hour: The plant recycles 4.9 kilograms.
- Next 7 hours: The plant recycles 7.8 kilograms per hour.
Next, we can calculate the total amount recycled over the 8-hour period:
- Total recycled during the first hour: \(4.9\) kg
- Total recycled during the next 7 hours: \[ 7 \text{ hours} \times 7.8 \text{ kg/hour} = 54.6 \text{ kg} \]
Now, add both amounts to get the total recycled over the 8 hours:
\[ \text{Total recycled} = 4.9 \text{ kg} + 54.6 \text{ kg} = 59.5 \text{ kg} \]
Since the recycling plant's function shows a constant rate after the first hour, the total function can be expressed as:
- From 0 to 1 hour: 4.9 kilograms
- From 1 to 8 hours: 4.9 + a linear increase as time progresses.
Thus, the function representation would be:
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At \( t = 0 \): \( 0 \text{ kg} \)
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At \( t = 1 \): \( 4.9 \text{ kg} \)
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From \( t = 1 \) to \( t = 8 \), the function continues to increase until:
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At \( t = 8 \): \( 59.5 \text{ kg} \)
Thus, the function is continuous, and the range of the total amount of recycled material from \( t = 0 \) to \( t = 8 \) hours will be from 0 kg to 59.5 kg.
The range of the function is then:
\[ \text{Range} = [4.9, 59.5] \]