Question
A rock climber is rappelling straight down a cliff. The table below shows the linear relationship between the time x (in minutes) since the rock climber began his descent and their height y (in meters) above the ground.
Time minutes
0
1
2
3
Height (meters)
16
12
8
4
Write a linear equation to represent the relationship between x and y.
Time minutes
0
1
2
3
Height (meters)
16
12
8
4
Write a linear equation to represent the relationship between x and y.
Answers
GPT-4o mini
To find the linear equation that represents the relationship between time \( x \) (in minutes) and height \( y \) (in meters), we can use the information given in the table.
The coordinates corresponding to the values provided are:
- \( (0, 16) \)
- \( (1, 12) \)
- \( (2, 8) \)
- \( (3, 4) \)
### Step 1: Determine the slope (m) of the line
The slope \( m \) of a line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the first two points \( (0, 16) \) and \( (1, 12) \):
\[
m = \frac{12 - 16}{1 - 0} = \frac{-4}{1} = -4
\]
### Step 2: Use the slope-intercept form of a line
The slope-intercept form of a line is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
We already found \( m = -4 \). To find \( b \), we can use one of the points, for example, the point \( (0, 16) \):
\[
16 = -4(0) + b \implies b = 16
\]
### Step 3: Write the linear equation
Now we can write the complete linear equation:
\[
y = -4x + 16
\]
This equation represents the linear relationship between the time \( x \) in minutes since the rock climber began their descent and their height \( y \) in meters above the ground.
The coordinates corresponding to the values provided are:
- \( (0, 16) \)
- \( (1, 12) \)
- \( (2, 8) \)
- \( (3, 4) \)
### Step 1: Determine the slope (m) of the line
The slope \( m \) of a line can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the first two points \( (0, 16) \) and \( (1, 12) \):
\[
m = \frac{12 - 16}{1 - 0} = \frac{-4}{1} = -4
\]
### Step 2: Use the slope-intercept form of a line
The slope-intercept form of a line is given by:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
We already found \( m = -4 \). To find \( b \), we can use one of the points, for example, the point \( (0, 16) \):
\[
16 = -4(0) + b \implies b = 16
\]
### Step 3: Write the linear equation
Now we can write the complete linear equation:
\[
y = -4x + 16
\]
This equation represents the linear relationship between the time \( x \) in minutes since the rock climber began their descent and their height \( y \) in meters above the ground.