Question

Caroline is planting flowers in her garden. She plants her first 7 flowers in 4 minutes. After 12 minutes, she has planted 21 different flowers.

Plot four data points and the line of best fit to model how long it takes Caroline to plant flowers.

Answer 1

To analyze Caroline's flower planting progress, we can establish a relationship between the number of flowers planted and the time taken. We will plot the available data based on Caroline's planting rate.

### Data Points

1. **First data point (0 flowers in 0 minutes)**:
- Time = 0 minutes, Flowers = 0
- (0, 0)

2. **Second data point (7 flowers in 4 minutes)**:
- Time = 4 minutes, Flowers = 7
- (4, 7)

3. **Third data point (21 flowers in 12 minutes)**:
- Time = 12 minutes, Flowers = 21
- (12, 21)

4. **Fourth data point (Total flowering rate calculation)**:
- To determine the average rate, we can calculate the time it takes to plant more flowers.
- Let's assume Caroline continues planting flowers at the same rate after 12 minutes. From the first two points, we observe that she planted 7 flowers in 4 minutes. The rate can be calculated as:
\[
\text{Rate} = \frac{7 \text{ flowers}}{4 \text{ minutes}} = 1.75 \text{ flowers per minute}
\]
- If we extrapolate this for another time interval, e.g., planting for another 4 minutes (16 minutes in total), we can estimate the total flowers planted:
- Additional flowers planted in 4 minutes = \(1.75 \text{ flowers/minute} \times 4 \text{ minutes} = 7 \text{ flowers}\)
- Total flowers after 16 minutes = 21 + 7 = 28 flowers.
- Thus we have another data point:
- (16, 28)

### Summary of Data Points
- (0, 0)
- (4, 7)
- (12, 21)
- (16, 28)

### Line of Best Fit
To find the line of best fit, we can compute the slope (m) based on the data points, particularly the last two points which are critical for creating a linear model.

1. **Slope (m) Calculation**:
- Using points (4, 7) and (12, 21):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{21 - 7}{12 - 4} = \frac{14}{8} = 1.75
\]

2. **Equation of the line**:
- The line has the slope \(1.75\) and goes through the y-intercept at (0, 0). Therefore, the equation of the line in slope-intercept form \(y = mx + b\) is:
\[
y = 1.75x
\]

### Plotting the Data Points and Line
Now we can plot the four data points and the line of best fit:

- **Data Points**:
- (0, 0)
- (4, 7)
- (12, 21)
- (16, 28)

- **Line of Best Fit**:
- The line can be represented graphically, showing a linear relationship between time (minutes) and the number of flowers planted.

### Conclusion
With the above data points and the line of best fit, we now have a simple, clear model of how Caroline's flower planting activity evolves over time. If you were to plot this in a graphing tool, you would see these points form a nearly straight line, reinforcing the consistency in her planting rate.