Question
Shane starts rock climbing at an elevation above sea level. He 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 Shane's elevation in meters. For each hour, his elevation increases at a constant rate. After 2 hours, Shane’s elevation is 630 meters. After 5 hours, his 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 Shane’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 =
This graph represents the linear relationship between the time in hours and Shane's elevation in meters. For each hour, his elevation increases at a constant rate. After 2 hours, Shane’s elevation is 630 meters. After 5 hours, his 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 Shane’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 =
Answers
GPT-4o mini
To find the slope \( m \) of the linear relationship between time in hours and Shane's elevation in meters, we can use the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points provided:
- Point 1: \( (2, 630) \)
- Point 2: \( (5, 1530) \)
Substituting these values into the slope formula:
\[
m = \frac{1530 - 630}{5 - 2} = \frac{900}{3} = 300
\]
So the slope \( m = 300 \). This means that Shane's elevation increases by 300 meters for each hour of climbing.
Now we can write the equation of the line in the form \( y = mx + b \). We already have \( m \) as 300. Next, we need to find the y-intercept \( b \). We can use one of the points to solve for \( b \). We'll use the point \( (2, 630) \):
\[
630 = 300(2) + b
\]
Solving for \( b \):
\[
630 = 600 + b \\
b = 630 - 600 \\
b = 30
\]
Now we have \( m = 300 \) and \( b = 30 \). Thus, the equation representing Shane's elevation \( y \) as a function of time \( x \) is:
\[
y = 300x + 30
\]
### Prediction
If we want to make a prediction, we can plug in a value for \( x \). For example, if we want to predict Shane's elevation after 4 hours:
\[
y = 300(4) + 30 \\
y = 1200 + 30 \\
y = 1230
\]
So after 4 hours, Shane's elevation is predicted to be 1230 meters.
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points provided:
- Point 1: \( (2, 630) \)
- Point 2: \( (5, 1530) \)
Substituting these values into the slope formula:
\[
m = \frac{1530 - 630}{5 - 2} = \frac{900}{3} = 300
\]
So the slope \( m = 300 \). This means that Shane's elevation increases by 300 meters for each hour of climbing.
Now we can write the equation of the line in the form \( y = mx + b \). We already have \( m \) as 300. Next, we need to find the y-intercept \( b \). We can use one of the points to solve for \( b \). We'll use the point \( (2, 630) \):
\[
630 = 300(2) + b
\]
Solving for \( b \):
\[
630 = 600 + b \\
b = 630 - 600 \\
b = 30
\]
Now we have \( m = 300 \) and \( b = 30 \). Thus, the equation representing Shane's elevation \( y \) as a function of time \( x \) is:
\[
y = 300x + 30
\]
### Prediction
If we want to make a prediction, we can plug in a value for \( x \). For example, if we want to predict Shane's elevation after 4 hours:
\[
y = 300(4) + 30 \\
y = 1200 + 30 \\
y = 1230
\]
So after 4 hours, Shane's elevation is predicted to be 1230 meters.