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 (1)
Calculate the rate of change between consecutive points: 2 − 1 4.75 − 2.5 = 2.25 .
Verify the rate of change between other points: 3 − 2 7 − 4.75 = 2.25 .
Confirm that the amount owed changes by 2.25 f ore v ery 1 mi l e . − C o n c l u d e t ha tt h er a t eo f c han g e i s 2.25 , so t h e an s w er i syes b ec a u se t h e am o u n t o w e d c han g es b y 2.25 e v ery t im e t h e mi l esc han g e b y 1 : \boxed{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 various numbers of miles traveled. We need to determine if the rate of change is 2.25. The rate of change can be found by calculating the change in the amount owed divided by the change in miles traveled between any two points in the table.
Calculating Rate of Change Let's calculate the rate of change between the first two points (1 mile, $2.5) and (2 miles, $4.75). The change in amount owed is $4.75 - $2.5 = 2.25. T h ec han g e inmi l es i s 2 − 1 = 1 mi l e . T h ere f ore , t h er a t eo f c han g e i s \frac{2.25}{1} = 2.25$ dollars per mile.
Verifying Rate of Change Now, let's calculate the rate of change between the second and third points (2 miles, $4.75) and (3 miles, $7). The change in amount owed is $7 - $4.75 = 2.25. T h ec han g e inmi l es i s 3 − 2 = 1 mi l e . T h ere f ore , t h er a t eo f c han g e i s \frac{2.25}{1} = 2.25$ dollars per mile.
Conclusion We can observe that for every 1 mile increase, the amount owed increases by $2.25. Therefore, the rate of change is indeed 2.25. The correct answer is 'Yes, because the amount owed changes by 2.25 every time the miles change by 1.'
Examples
Understanding rate of change is crucial in everyday scenarios like calculating the cost of a taxi ride. For instance, if a taxi charges a base fare plus a rate per mile, knowing the rate of change helps you estimate the total cost of your journey. Suppose the base fare is $1 and the rate per mile is $2.25. If you travel 5 miles, the total cost would be $1 + ($2.25 * 5) = $12.25. This concept extends to budgeting, understanding utility bills, and making informed financial decisions.
