Question

Betsy starts rock climbing at an elevation above sea level. She climbs up at a constant rate per hour. 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 hours and Betsy's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Betsy’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 Betsy’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)
y= mx + b

Slope =

Answer 1

To determine Betsy's slope, we first need to understand that the slope (m) of a line is defined as the change in the y-values (elevation) divided by the change in the x-values (time). We can use the two points given: (2, 630) and (5, 1530).

The formula for calculating the slope is given by:

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

Where:
- \((x_1, y_1) = (2, 630)\)
- \((x_2, y_2) = (5, 1530)\)

Now we can substitute the values into the formula:

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

Calculating the differences:

\[
m = \frac{900}{3} = 300
\]

So the slope, \(m\), is 300 meters per hour. This means Betsy's elevation increases by 300 meters for every hour she climbs.

Next, we need to write the equation of the line in the form \(y = mx + b\). We already have \(m = 300\).

Now we can find \(b\), the y-intercept, using one of the points. Let's use the point (2, 630):

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

Now we have both \(m\) and \(b\). Therefore, the equation that represents Betsy's situation is:

\[
y = 300x + 30
\]

Where:
- \(y\) is the elevation in meters,
- \(x\) is the time in hours.

Now, let's make a prediction using this equation. For example, if Betsy climbs for 4 hours, we can find her elevation by substituting \(x = 4\) into the equation:

\[
y = 300(4) + 30
\]
\[
y = 1200 + 30 = 1230
\]

Thus, after 4 hours, Betsy's elevation would be 1230 meters.

In summary:
- The slope (change in elevation per hour) is 300 m/h.
- The equation of the line is \(y = 300x + 30\).
- Prediction after 4 hours: 1230 meters.