Betsy starts hiking at an elevation above sea level. She hikes up at a constant rate per minute. 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 minutes and Betsy's elevation in meters. For each minute, her elevation increases at a constant rate. After 4 minutes, Betsy’s elevation is 140 meters. After 8 minutes, her elevation is 260 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 minute?
x
y
(4, 140)(8, 260)
0
2
4
6
8
10
12
14
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
380
400
Time (minutes)
Elevation (meters)
y= mx + b
Slope =
To find the slope, we can use the formula:
Slope = (change in y)/(change in x)
So, the slope is:
Slope = (260 - 140)/(8 - 4)
Slope = 120/4
Slope = 30
Therefore, the slope of the line representing Betsy's elevation is 30 meters per minute.
Now, we can use the point-slope form of the equation of a line to represent this situation.
The point-slope form of the equation of a line is: y - y1 = m(x - x1)
Using the point (4, 140) which is on the line, we get:
y - 140 = 30(x - 4)
Now, we can simplify this to get the equation in the form y = mx + b:
y = 30x - 120 + 140
y = 30x + 20
Therefore, the equation that represents Betsy's elevation above sea level is y = 30x + 20.
To make a prediction, we can substitute any value of x into the equation to find the corresponding elevation. For example, if Betsy hikes for 10 minutes, her elevation would be:
y = 30(10) + 20
y = 300 + 20
y = 320
Therefore, after 10 minutes, Betsy's elevation would be 320 meters above sea level.