Question

Claire is a manager at a toy packaging company. The company packs 80 boxes of toys every hour for the first 3 hours of the day. They stop packaging for the next 2 hours in order to carry out a training session. Then, for the next 4 hours, they pack 20 boxes of toys every hour.

Which graph models the piecewise function for the given situation?

A.
The piecewise function represents the number of hours on the x-axis and the number of boxes on the y-axis. The function starts from (0, 0), (3, 240), (5, 240), and (9, 280).
B.
The piecewise function represents the number of hours on the x- axis and the number of boxes on the y- axis. The function starts from (0, 0), (3, 240), (5, 240), and (9, 320).
C.
The piecewise function represents the number of hours on the x-axis and the number of boxes on the y-axis. The function starts from (0, 0), (3, 240), (7, 320), and (9, 320).
D.
The piecewise function represents the number of hours on the x-axis and the number of boxes on the y- axis. The function starts from (0, 0), (4, 60), (6, 60), and (9, 320).

Answer 1

To analyze the situation described, we need to calculate the number of boxes packaged over the specified time periods.

1. For the **first 3 hours**: The company packs 80 boxes per hour.
- Boxes packed in first 3 hours = \(80 \text{ boxes/hour} \times 3 \text{ hours} = 240 \text{ boxes}\)

2. For the **next 2 hours**: There is no packaging during the training session.
- Boxes packed = 0 during this period.

3. For the **next 4 hours**: The company packs 20 boxes per hour.
- Boxes packed in the next 4 hours = \(20 \text{ boxes/hour} \times 4 \text{ hours} = 80 \text{ boxes}\)

Now we can sum the total number of boxes packed by the end of each time frame:

- After 3 hours: 240 boxes
- After 5 hours (3 hours of packing + 2 hours of training): 240 boxes (no change)
- After 9 hours (5 hours from the previous period + 4 hours of packing): \(240 + 80 = 320 \text{ boxes}\)

Now we can determine the points for our piecewise function:

- Starting point: (0, 0) (at the start, no boxes packed yet)
- After 3 hours: (3, 240)
- After 5 hours: (5, 240) (still 240 boxes due to break)
- After 9 hours: (9, 320)

Based on the calculations, the piecewise function has points (0, 0), (3, 240), (5, 240), and (9, 320).

Now we can compare our results to the options provided:

- Option A: (0, 0), (3, 240), (5, 240), (9, 280) – Incorrect, last point is wrong.
- Option B: (0, 0), (3, 240), (5, 240), (9, 320) – Correct.
- Option C: (0, 0), (3, 240), (7, 320), (9, 320) – Incorrect, misrepresents the timing.
- Option D: (0, 0), (4, 60), (6, 60), (9, 320) – Incorrect points.

Therefore, the correct answer is **B**.