To find the slope \( m \) of the line representing Cecilia's elevation over time, we can use the two points provided: (4, 140) and (8, 260).
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in our points:
- \( (x_1, y_1) = (4, 140) \)
- \( (x_2, y_2) = (8, 260) \)
We calculate the slope as follows:
\[ m = \frac{260 - 140}{8 - 4} = \frac{120}{4} = 30 \]
So, Cecilia's change in elevation for each minute is \( 30 \) meters per minute.
Next, we can write the equation of the line in the slope-intercept form \( y = mx + b \). We already found that \( m = 30 \).
To find the y-intercept \( b \), we can use one of the points. Let's use \( (4, 140) \):
\[ 140 = 30(4) + b \] \[ 140 = 120 + b \] \[ b = 140 - 120 = 20 \]
Now we can write the equation of the line:
\[ y = 30x + 20 \]
Where \( y \) is the elevation in meters, and \( x \) is the time in minutes.
Prediction:
To make a prediction, we can plug in any value of \( x \) into the equation. For instance, if we want to predict Cecilia's elevation after 10 minutes:
\[ y = 30(10) + 20 = 300 + 20 = 320 \]
Thus, after 10 minutes, Cecilia's elevation is predicted to be 320 meters.