Slope in Real-World Problems Quick Check
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Question
Use the table to answer the question.
Height Above Canyon Floor Time (minutes)
360 0
280 3
160 10
80 14
Which statement best interprets the rate of change of the linear model shown in the table?
(1 point)
Responses
The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute.
The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute.
The elevation of a hiker who is hiking down to a canyon floor changes at a rate of −20 feet per minute.
The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.
The elevation of a hiker who is hiking up from canyon floor changes at a rate of −20 feet per minute.
The elevation of a hiker who is hiking up from canyon floor changes at a rate of negative 20 feet per minute.
The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute.
The correct answer is:
The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute.
The statement that best interprets the rate of change of the linear model shown in the table is:
"The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute."
To determine the rate of change from the given table, we need to calculate the slope of the linear model. The slope represents the change in the dependent variable (elevation) for every one unit increase in the independent variable (time).
To calculate the slope, we can choose two data points from the table. Let's choose the first and last data points: (360, 0) and (80, 14).
The formula for slope is:
slope = (change in y) / (change in x)
In this case, the change in y is the change in elevation, and the change in x is the change in time.
change in y = (elevation at second point - elevation at first point) = 0 - 14 = -14
change in x = (time at second point - time at first point) = 0 - 14 = -14
slope = (-14) / (-14) = 1
The slope of the linear model is 1.
Now let's interpret the statement that best represents this rate of change.
Option 1: The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute.
Since the slope we calculated is 1, this statement is not correct. The correct interpretation should reflect a rate of change of 1, not 20 feet per minute.
Option 2: The elevation of a hiker who is hiking down to a canyon floor changes at a rate of -20 feet per minute.
This statement reflects a negative rate of change, but the magnitude is incorrect. The correct rate of change is 1, not 20 feet per minute.
Option 3: The elevation of a hiker who is hiking up from canyon floor changes at a rate of -20 feet per minute.
Again, this statement reflects a negative rate of change, but the magnitude is incorrect. The correct rate of change is 1, not -20 feet per minute.
Option 4: The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute.
This statement does not reflect the correct direction of the rate of change. The correct interpretation should reflect the hiker moving up from the canyon floor, not down.
Therefore, the correct answer is:
The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 1 foot per minute.
Note: We did not have data points to calculate the exact rate of change as the units for time are minutes, not one unit of time. However, since the slope is 1, we can interpret it as a rate of 1 foot per minute.