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

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?

Number of minutes
4
6
12
14

number of items packed

20
20
44
50

please double check

5 answers

To analyze the rate of change of the number of items packed with respect to time, we can calculate the number of items packed per minute over the different time intervals given.

Let's look at the data provided and calculate the rate of change:

  • For 4 minutes: 20 items packed
  • For 6 minutes: 20 items packed
  • For 12 minutes: 44 items packed
  • For 14 minutes: 50 items packed

Now let’s check the intervals:

  1. From 4 to 6 minutes:

    • Items packed = 20 - 20 = 0
    • Time interval = 6 - 4 = 2 minutes
    • Rate of change = 0 items / 2 minutes = 0 items per minute

  2. From 6 to 12 minutes:

    • Items packed = 44 - 20 = 24
    • Time interval = 12 - 6 = 6 minutes
    • Rate of change = 24 items / 6 minutes = 4 items per minute

  3. From 12 to 14 minutes:

    • Items packed = 50 - 44 = 6
    • Time interval = 14 - 12 = 2 minutes
    • Rate of change = 6 items / 2 minutes = 3 items per minute

Now summarizing the rates of change for each interval:

  • From 4 to 6 minutes: \(0\) items per minute
  • From 6 to 12 minutes: \(4\) items per minute
  • From 12 to 14 minutes: \(3\) items per minute

Based on the calculations, the rate of change of the number of items packed in boxes varies over time, indicating that the worker's packing efficiency is not constant.

In conclusion, the statement describing the rate of change is that the rate of packing items varies depending on the time intervals, showing an increase from 0 to 4 items per minute in the earlier intervals, and then a decrease to 3 items per minute later on.

The altitude of an airplane is changing at a constant rate. The table shows the linear relationship between
y, the altitude of the airplane in feet, and
x, the time in minutes.

Complete the statement that describes the rate of change of the altitude of the airplane with respect to time.

Times (x)
1.5
4.25
3
Altitude, y (feet)
28,000
24,500
19,000

charts to fill

what number
The altitude of the airplane
increase or decrease

fill this blank with a number

fill this blank with the rate at a rate of
1,000 per miniute
2,000 feet per miniute
3,000 feet per minitue
4,000 feet per minute

To analyze the rate of change of the altitude of the airplane with respect to time, we will look at the given data points and calculate the altitude changes over the time intervals.

Given Data

  • At \(x = 1.5\) minutes, \(y = 28,000\) feet
  • At \(x = 4.25\) minutes, \(y = 24,500\) feet
  • At \(x = 3\) minutes, \(y = 19,000\) feet

Finding the Rate of Change

We'll calculate the altitude change between each pair of times.

  1. From 1.5 to 4.25 minutes:

    • Change in altitude: \(24,500 - 28,000 = -3,500\) feet
    • Change in time: \(4.25 - 1.5 = 2.75\) minutes
    • Rate of change: \(-3,500 \text{ feet} / 2.75 \text{ minutes} \approx -1,272.73\) feet per minute

  2. From 1.5 to 3 minutes:

    • Change in altitude: \(19,000 - 28,000 = -9,000\) feet
    • Change in time: \(3 - 1.5 = 1.5\) minutes
    • Rate of change: \(-9,000 \text{ feet} / 1.5 \text{ minutes} = -6,000\) feet per minute

  3. From 3 to 4.25 minutes:

    • Change in altitude: \(24,500 - 19,000 = 5,500\) feet
    • Change in time: \(4.25 - 3 = 1.25\) minutes
    • Rate of change: \(5,500 \text{ feet} / 1.25 \text{ minutes} = 4,400\) feet per minute

Summary of Rates:

  • From 1.5 to 4.25 mins: approximately -1,273 feet/minute
  • From 1.5 to 3 mins: -6,000 feet/minute
  • From 3 to 4.25 mins: approximately 4,400 feet/minute

From this analysis, we can see that the altitude of the airplane is decreasing during the first two time intervals and then increasing in the last interval.

Filling in the Blanks

Given that the overall pattern shows a significant decrease in altitude at the lowest of the intervals you provided (specifically the largest negative rate calculated):

  1. The altitude of the airplane:

    • Decrease

  2. Fill this blank with a number:

    • The most significant decrease in altitude occurs at around 6,000 feet per minute from 1.5 to 3 minutes.

  3. Fill this blank with the rate of:

    • 6,000 feet per minute

Therefore, the completed statement would be:

  • The altitude of the airplane decrease by 6,000 feet per minute.
The altitude of an airplane is changing at a constant rate. The table shows the linear relationship between
y, the altitude of the airplane in feet, and
x, the time in minutes.

Complete the statement that describes the rate of change of the altitude of the airplane with respect to time.

Times (x)
1.5
4.25
3
Altitude, y (feet)
28,000
24,500
19,000

charts to fill

what number
The altitude of the airplane
increase or decrease

fill this blank with a number

fill this blank with the rate at a rate of
1,000 per miniute
2,000 feet per miniute
3,000 feet per minitue
4,000 feet per minute

pick one of these answers and solve for X and Y

To determine the rate of change of the altitude of the airplane, we will calculate the altitude changes at various time intervals based on the given data.

Given Data

  • At \(x = 1.5\) minutes, \(y = 28,000\) feet
  • At \(x = 3\) minutes, \(y = 19,000\) feet
  • At \(x = 4.25\) minutes, \(y = 24,500\) feet

Finding the Rate of Change

  1. From 1.5 to 3 minutes:

    • Change in altitude: \[ y_2 - y_1 = 19,000 - 28,000 = -9,000 \text{ feet} \]
    • Change in time: \[ x_2 - x_1 = 3 - 1.5 = 1.5 \text{ minutes} \]
    • Rate of change: \[ \text{Rate} = \frac{-9,000 \text{ feet}}{1.5 \text{ minutes}} = -6,000 \text{ feet per minute} \]

  2. From 3 to 4.25 minutes:

    • Change in altitude: \[ y_3 - y_2 = 24,500 - 19,000 = 5,500 \text{ feet} \]
    • Change in time: \[ x_3 - x_2 = 4.25 - 3 = 1.25 \text{ minutes} \]
    • Rate of change: \[ \text{Rate} = \frac{5,500 \text{ feet}}{1.25 \text{ minutes}} = 4,400 \text{ feet per minute} \]

  3. From 1.5 to 4.25 minutes (overall change):

    • Change in altitude: \[ y(4.25) - y(1.5) = 24,500 - 28,000 = -3,500 \text{ feet} \]
    • Change in time: \[ x(4.25) - x(1.5) = 4.25 - 1.5 = 2.75 \text{ minutes} \]
    • Rate of change: \[ \text{Rate} = \frac{-3,500 \text{ feet}}{2.75 \text{ minutes}} \approx -1,272.73 \text{ feet per minute} \]

Summary of Findings:

  1. From 1.5 to 3 minutes: -6,000 feet per minute (decrease)
  2. From 3 to 4.25 minutes: +4,400 feet per minute (increase)
  3. Overall from 1.5 to 4.25 minutes: approximately -1,272.73 feet per minute (decrease)

Filling in the Statements

Now we can fill in the blanks based on your question:

  1. The altitude of the airplane:

    • Decrease

  2. Fill this blank with a number:

    • Based on the first interval of time, the most significant decrease noted is 6,000 feet per minute.

  3. Fill this blank with the rate of:

    • The closest option to the calculated rates (1,000, 2,000, 3,000, or 4,000 feet per minute) is 4,000 feet per minute, assuming we only consider the positive context from the increase in altitude from the last interval.

Thus, we can summarize with the statements:

  • The altitude of the airplane decreases.
  • Fill this blank with a number: 6,000.
  • Fill this blank with the rate at a rate of: 4,000 feet per minute.

Please note that none of the provided options perfectly match the calculations, but we've done our best to align them with the given choices.