a worker is packing items in a boxes. The table shows the linear relationship between the number of items the worker packed in boxes after different amounts of time.

Number of minutes: 4, 6, 12, 14
Number of items packed: 20, 26, 44, 50
which statement describes the rate of change of the number of items the worker packed in boxes with respect to the number of minutes the worker has been packing items in boxes?

1 answer

To find the rate of change of the number of items packed with respect to the number of minutes, we can calculate the slope of the linear relationship using the provided data points.

The formula for the slope (rate of change) \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

From the data provided, we can choose any two points. For instance, we'll calculate the rate of change using the points (4 minutes, 20 items) and (14 minutes, 50 items).

Letting:

  • \( (x_1, y_1) = (4, 20) \)
  • \( (x_2, y_2) = (14, 50) \)

Plugging in the values:

\[ m = \frac{50 - 20}{14 - 4} = \frac{30}{10} = 3 \]

The rate of change is 3 items per minute. This means that for every additional minute the worker packs items, they pack an average of 3 more items.

Thus, the statement that describes the rate of change is:

The worker packs 3 items for each minute spent packing.