Question
Tamia 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 Tamia's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Tamia’s elevation is 400 meters. After 4 hours, her elevation is 720 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 Tamia’s change in elevation for each hour?
x
y
(2, 400)(4, 720)
0
1
2
3
4
5
6
7
0
80
160
240
320
400
480
560
640
720
800
880
960
1,040
1,120
1,200
1,280
1,360
Time (hours)
Elevation (meters)
y= mx + b
Slope =
This graph represents the linear relationship between the time in hours and Tamia's elevation in meters. For each hour, her elevation increases at a constant rate. After 2 hours, Tamia’s elevation is 400 meters. After 4 hours, her elevation is 720 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 Tamia’s change in elevation for each hour?
x
y
(2, 400)(4, 720)
0
1
2
3
4
5
6
7
0
80
160
240
320
400
480
560
640
720
800
880
960
1,040
1,120
1,200
1,280
1,360
Time (hours)
Elevation (meters)
y= mx + b
Slope =
Answers
GPT-4o mini
To find the slope (m) of the line representing Tamia's elevation in meters (y) as a function of time in hours (x), we can use the two points provided: (2, 400) and (4, 720).
The formula for slope (m) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points:
- \( (x_1, y_1) = (2, 400) \)
- \( (x_2, y_2) = (4, 720) \)
Substituting the values:
\[
m = \frac{720 - 400}{4 - 2} = \frac{320}{2} = 160
\]
So the slope \( m = 160 \). This means Tamia's elevation increases by 160 meters for each hour she climbs.
Next, we need to write the equation in the form \( y = mx + b \). We have \( m = 160 \), but we need to find \( b \), the y-intercept, which represents the initial elevation at time \( x = 0 \).
We can use one of the points to solve for \( b \). Let's use the point (2, 400):
\[
y = mx + b
\]
\[
400 = 160(2) + b
\]
\[
400 = 320 + b
\]
\[
b = 400 - 320 = 80
\]
Now we have \( b = 80 \).
The equation representing Tamia's elevation is:
\[
y = 160x + 80
\]
### Prediction
To make a prediction, you can input a value for \( x \) (the time in hours) into the equation. For example, if you want to find out Tamia's elevation after 6 hours:
\[
y = 160(6) + 80
\]
\[
y = 960 + 80
\]
\[
y = 1040
\]
Tamia's elevation after 6 hours will be **1040 meters**.
The formula for slope (m) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points:
- \( (x_1, y_1) = (2, 400) \)
- \( (x_2, y_2) = (4, 720) \)
Substituting the values:
\[
m = \frac{720 - 400}{4 - 2} = \frac{320}{2} = 160
\]
So the slope \( m = 160 \). This means Tamia's elevation increases by 160 meters for each hour she climbs.
Next, we need to write the equation in the form \( y = mx + b \). We have \( m = 160 \), but we need to find \( b \), the y-intercept, which represents the initial elevation at time \( x = 0 \).
We can use one of the points to solve for \( b \). Let's use the point (2, 400):
\[
y = mx + b
\]
\[
400 = 160(2) + b
\]
\[
400 = 320 + b
\]
\[
b = 400 - 320 = 80
\]
Now we have \( b = 80 \).
The equation representing Tamia's elevation is:
\[
y = 160x + 80
\]
### Prediction
To make a prediction, you can input a value for \( x \) (the time in hours) into the equation. For example, if you want to find out Tamia's elevation after 6 hours:
\[
y = 160(6) + 80
\]
\[
y = 960 + 80
\]
\[
y = 1040
\]
Tamia's elevation after 6 hours will be **1040 meters**.