Question

This graph represents the linear relationship between the time in hours and Irene's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Irene’s elevation is 630 meters. After 5 hours, her elevation is 1,530 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 Irene’s change in elevation for each hour?
x
y
(2, 630)(5, 1,530)
0
1
2
3
4
5
6
7
8
9
10
11
12
0
180
360
540
720
900
1,080
1,260
1,440
1,620
1,800
1,980
2,160
2,340
2,520
2,700
Time (hours)
Elevation (meters)

Answer 1

To find the slope \(m\) of the line representing Irene's elevation as a function of time, we can use the formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

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

Given the points from the graph:
- Point 1: \( (2, 630) \)
- Point 2: \( (5, 1530) \)

Let's identify \( (x_1, y_1) = (2, 630) \) and \( (x_2, y_2) = (5, 1530) \).

Now calculate the slope \(m\):

\[
m = \frac{1530 - 630}{5 - 2} = \frac{900}{3} = 300
\]

Thus, the slope \(m\) is \(300\). This means Irene's elevation increases by **300 meters for each hour.**

To write the equation in the form \(y = mx + b\), we will also need to find the y-intercept \(b\). We can use one of the points to find \(b\).

Using the point \( (2, 630) \):

\[
630 = 300(2) + b
\]
\[
630 = 600 + b
\]
\[
b = 630 - 600 = 30
\]

Now we have both \(m\) and \(b\):

- Slope \(m = 300\)
- Y-intercept \(b = 30\)

So the equation that represents the relationship between time in hours and Irene's elevation in meters is:

\[
y = 300x + 30
\]