Juno is taking a taxi. The table represents a linear function and shows the amount she owed after various numbers of miles traveled. Is the rate of change 2.25?
| Miles | Amount Owed (dollars) |
| ----- | --------------------- |
| 1 | 2.5 |
| 2 | 4.75 |
| 3 | 7 |
| 4 | 9.25 |
| 5 | 11.5 |
A. Yes, because the amount owed changes by 1 every time the miles change by 2.25.
B. Yes, because the amount owed changes by 2.25 every time the miles change by 1.
C. No, because the amount owed does not change by 1 every time the miles change by 2.25.
D. No, because the amount owed does not change by 2.25 every time the miles change by 1.
Answer (2)
Calculate the rate of change between the first two points: 2 − 1 4.75 − 2.5 = 2.25 .
Verify the rate of change between other pairs of points to ensure it's constant.
Confirm that the rate of change is consistently 2.25.
Conclude that the rate of change is 2.25, meaning the amount owed changes by $2.25 for every mile traveled, so the answer is Yes, because the amount owed changes by 2.25 every time the miles change by 1.
Explanation
Understanding the Problem We are given a table that shows the amount Juno owed for a taxi after traveling a certain number of miles. We need to determine if the rate of change is 2.25. The rate of change represents how much the amount owed changes for each mile traveled.
Calculating the Rate of Change To find the rate of change, we can calculate the change in the amount owed divided by the change in miles traveled between any two points in the table. Let's calculate the rate of change between the first two points (1 mile, $2.5) and (2 miles, $4.75).
Determining the Rate of Change The rate of change is calculated as follows: Change in miles Change in amount owed = 2 − 1 4.75 − 2.5 = 1 2.25 = 2.25 So, the amount owed increases by $2.25 for every 1 mile traveled.
Verifying the Rate of Change We can verify this by calculating the rate of change between other pairs of points. For example, between (2 miles, $4.75) and (3 miles, 7 ) : 3 − 2 7 − 4.75 = 1 2.25 = 2.25 $
Between (3 miles, $7) and (4 miles, 9.25 ) : 4 − 3 9.25 − 7 = 1 2.25 = 2.25 $
Between (4 miles, $9.25) and (5 miles, 11.5 ) : 5 − 4 11.5 − 9.25 = 1 2.25 = 2.25 $
Since the rate of change is consistently 2.25, the statement is correct.
Final Answer The rate of change is indeed 2.25, meaning that for every 1 mile Juno travels, the amount she owes increases by $2.25. Therefore, the correct answer is: Yes, because the amount owed changes by 2.25 every time the miles change by 1.
Examples
Imagine you're tracking the cost of a phone plan. If the plan charges a fixed amount per gigabyte of data used, the rate of change would represent the cost per gigabyte. For instance, if the cost increases by $5 for every 1 GB of data, the rate of change is $5/GB. Understanding this rate helps you predict your monthly bill based on your data usage, allowing you to budget effectively and avoid unexpected charges. This concept applies to various real-life scenarios where costs increase linearly with usage.
The rate of change for Juno's taxi fare is indeed 2.25, as confirmed by calculations between the points on the table. This means the amount owed increases by $2.25 for every mile she travels. Thus, the answer is B: Yes, because the amount owed changes by 2.25 every time the miles change by 1.
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