Question

Cecilia starts hiking at an elevation above sea level. She hikes up at a constant rate per minute. Write an equation to represent this situation. Then use the equation to make a prediction.
This graph represents the linear relationship between the time in minutes and Cecilia's elevation in meters. For each minute, her elevation increases at a constant rate. After 4 minutes, Cecilia’s elevation is 140 meters. After 8 minutes, her elevation is 260 meters. Let’s write an equation to represent this line in the form y=mx+b. Start by finding m, or the slope. What is the slope of this line? In other words, what is Cecilia’s change in elevation for each minute?
x
y
(4, 140)(8, 260)
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Time (minutes)
Elevation (meters)
y= mx + b

Slope =

Answer 1

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**.