Question
Use the table to answer the question.
Applying the provided information to analyze the rate of decline in height above the canyon floor over time.
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 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 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 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.
Applying the provided information to analyze the rate of decline in height above the canyon floor over time.
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 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 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 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.
Answers
GPT-4o mini
To determine the rate of change in height above the canyon floor based on the provided data, we can analyze the changes in height over the changes in time.
From the data:
- At time 0 minutes, the height is 360 feet.
- At time 3 minutes, the height is 280 feet.
- At time 10 minutes, the height is 160 feet.
- At time 14 minutes, the height is 80 feet.
We can calculate the change in height between consecutive time intervals:
1. From 0 to 3 minutes: \( 360 - 280 = 80 \) feet decrease
2. From 3 to 10 minutes: \( 280 - 160 = 120 \) feet decrease
3. From 10 to 14 minutes: \( 160 - 80 = 80 \) feet decrease
Next, we will calculate the time intervals for each of these changes:
1. From 0 to 3 minutes: 3 minutes
2. From 3 to 10 minutes: 7 minutes
3. From 10 to 14 minutes: 4 minutes
Now we can find the average rate of change over the entire duration from 0 to 14 minutes:
- The total change in height from 360 to 80 feet is \( 360 - 80 = 280 \) feet.
- The total time is \( 14 - 0 = 14 \) minutes.
Therefore, the average rate of change is:
\[
\text{Rate} = \frac{\text{Change in height}}{\text{Change in time}} = \frac{280 \text{ feet}}{14 \text{ minutes}} = 20 \text{ feet per minute (downward)}
\]
Since the height is decreasing over time, the appropriate interpretation of this rate of change is that it is negative. Thus, the best interpretation of the rate of change is:
**The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.**
From the data:
- At time 0 minutes, the height is 360 feet.
- At time 3 minutes, the height is 280 feet.
- At time 10 minutes, the height is 160 feet.
- At time 14 minutes, the height is 80 feet.
We can calculate the change in height between consecutive time intervals:
1. From 0 to 3 minutes: \( 360 - 280 = 80 \) feet decrease
2. From 3 to 10 minutes: \( 280 - 160 = 120 \) feet decrease
3. From 10 to 14 minutes: \( 160 - 80 = 80 \) feet decrease
Next, we will calculate the time intervals for each of these changes:
1. From 0 to 3 minutes: 3 minutes
2. From 3 to 10 minutes: 7 minutes
3. From 10 to 14 minutes: 4 minutes
Now we can find the average rate of change over the entire duration from 0 to 14 minutes:
- The total change in height from 360 to 80 feet is \( 360 - 80 = 280 \) feet.
- The total time is \( 14 - 0 = 14 \) minutes.
Therefore, the average rate of change is:
\[
\text{Rate} = \frac{\text{Change in height}}{\text{Change in time}} = \frac{280 \text{ feet}}{14 \text{ minutes}} = 20 \text{ feet per minute (downward)}
\]
Since the height is decreasing over time, the appropriate interpretation of this rate of change is that it is negative. Thus, the best interpretation of the rate of change is:
**The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute.**