Question

A recycling plant recycled 4.9 kilograms of paper and cardboard for the first hour and 7.8 kilograms per hour afterwards. The plant runs for 8 hours. What is the range of the function?

Answer 1

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.

1. **First hour:** The plant recycles 4.9 kilograms.
2. **Next 7 hours:** The plant recycles 7.8 kilograms per hour.

Next, we can calculate the total amount recycled over the 8-hour period:

1. Total recycled during the first hour: \(4.9\) kg
2. 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:
- At \( t = 0 \): \( 0 \text{ kg} \)
- At \( t = 1 \): \( 4.9 \text{ kg} \)
- From \( t = 1 \) to \( t = 8 \), the function continues to increase until:

- 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:

\[
\text{Range} = [4.9, 59.5]
\]