To find the equation that models the fine schedule for overdue books, we can observe the relationship between the number of days overdue and the amount of the fine in cents.
From the data in the fine schedule:
- After 0 days, the fine is 0 cents.
- After 2 days, the fine is 5 cents.
- After 4 days, the fine is 10 cents.
- After 6 days, the fine is 15 cents.
- After 8 days, the fine is 20 cents.
- After 10 days, the fine is 25 cents.
We can summarize this data as:
\[ \begin{align*} (0, 0) \ (2, 5) \ (4, 10) \ (6, 15) \ (8, 20) \ (10, 25) \ \end{align*} \]
We can see that the amount of the fine (y) increases as the number of days overdue (x) increases. In fact, for every 2 days overdue, the fine increases by 5 cents. This indicates a linear relationship where the slope (rate of increase of the fine) is:
\[ \text{slope} = \frac{5 \text{ cents}}{2 \text{ days}} = \frac{5}{2} \text{ cents per day} \]
To formulate the linear equation, we can start with the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
From the data:
- The y-intercept \( b = 0 \) (the fine is 0 when days are 0).
Thus, the equation can be written as:
\[ y = \frac{5}{2} x \]
Testing the provided options against this result:
- y = five-halves x, where x is the cost in cents for a book that is y days overdue - Incorrect.
- y = five-halves x, where y is the cost in cents for a book that is x days overdue - Correct.
- y = 5 x, where x is the cost in cents for a book that is y days overdue - Incorrect.
- y = 5 x, where y is the cost in cents for a book that is x days overdue - Incorrect.
The best equation that models the fine schedule for overdue books is:
y = five-halves x, where y is the cost in cents for a book that is x days overdue.
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