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Sagot :
To determine the output with a constant rate of change and identify the linear relationship, we need to analyze the given data for day, average speed, and distance.
### Step-by-Step Solution
1. Rate of Change Analysis:
Let's first find the rate of change (also known as slope) for both distance and average speed with respect to days.
Rate of Change for Distance:
[tex]\[ \text{Rate of Change (Distance)} = \frac{\text{Change in Distance}}{\text{Change in Days}} = \frac{D_{\text{final}} - D_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} = \frac{1155 - 495}{7 - 3} = \frac{660}{4} = 165 \text{ miles/day} \][/tex]
The distance traveled increases by 165 miles each day.
Rate of Change for Average Speed:
[tex]\[ \text{Rate of Change (Speed)} = \frac{\text{Change in Speed}}{\text{Change in Days}} = \frac{S_{\text{final}} - S_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} = \frac{68 - 55}{7 - 3} = \frac{13}{4} = 3.25 \text{ mph/day} \][/tex]
The speed increases by 3.25 mph each day.
2. Constant Rate of Change:
- We observe that the distance changes by a constant rate (165 miles/day).
- We also observe the speed changes by 3.25 mph each day, but it’s not consistent for all intervals.
Thus, the output that has a constant rate of change with respect to days is distance.
3. Determine the Linear Relationship:
Since the distance increases consistently by the same amount each day, we say there is a linear relationship between the day and the distance traveled.
4. Conclusion:
- Constant Rate of Change: The constant rate of change is the rate at which the output changes with respect to the input. For distance, it is 165 miles/day.
- Linear Function Relationship: The relationship that represents a linear function is the one where the output changes at a constant rate with respect to the input. In this case, it is the relationship between day and distance.
### Summary of the Answers:
- Output with a constant rate of change: Distance
- Constant rate of change: 165 miles/day
- Linear function relationship: day to distance
Thus, the completed answer is:
1. Given that the days are the input, which output has a constant rate of change?
[tex]\[ \text{distance} \][/tex]
2. What is the constant rate of change?
[tex]\[ 165 \text{ miles/day} \][/tex]
3. Which relationship represents a linear function?
[tex]\[ \text{day to distance} \][/tex]
### Step-by-Step Solution
1. Rate of Change Analysis:
Let's first find the rate of change (also known as slope) for both distance and average speed with respect to days.
Rate of Change for Distance:
[tex]\[ \text{Rate of Change (Distance)} = \frac{\text{Change in Distance}}{\text{Change in Days}} = \frac{D_{\text{final}} - D_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} = \frac{1155 - 495}{7 - 3} = \frac{660}{4} = 165 \text{ miles/day} \][/tex]
The distance traveled increases by 165 miles each day.
Rate of Change for Average Speed:
[tex]\[ \text{Rate of Change (Speed)} = \frac{\text{Change in Speed}}{\text{Change in Days}} = \frac{S_{\text{final}} - S_{\text{initial}}}{t_{\text{final}} - t_{\text{initial}}} = \frac{68 - 55}{7 - 3} = \frac{13}{4} = 3.25 \text{ mph/day} \][/tex]
The speed increases by 3.25 mph each day.
2. Constant Rate of Change:
- We observe that the distance changes by a constant rate (165 miles/day).
- We also observe the speed changes by 3.25 mph each day, but it’s not consistent for all intervals.
Thus, the output that has a constant rate of change with respect to days is distance.
3. Determine the Linear Relationship:
Since the distance increases consistently by the same amount each day, we say there is a linear relationship between the day and the distance traveled.
4. Conclusion:
- Constant Rate of Change: The constant rate of change is the rate at which the output changes with respect to the input. For distance, it is 165 miles/day.
- Linear Function Relationship: The relationship that represents a linear function is the one where the output changes at a constant rate with respect to the input. In this case, it is the relationship between day and distance.
### Summary of the Answers:
- Output with a constant rate of change: Distance
- Constant rate of change: 165 miles/day
- Linear function relationship: day to distance
Thus, the completed answer is:
1. Given that the days are the input, which output has a constant rate of change?
[tex]\[ \text{distance} \][/tex]
2. What is the constant rate of change?
[tex]\[ 165 \text{ miles/day} \][/tex]
3. Which relationship represents a linear function?
[tex]\[ \text{day to distance} \][/tex]
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