To determine which equation best models the fine schedule for overdue books, we will analyze the given data and check which equation fits this data.
Given data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Days overdue} & \text{Amount of fine (Cents)} \\
\hline
0 & 0 \\
\hline
2 & 5 \\
\hline
4 & 10 \\
\hline
6 & 15 \\
\hline
8 & 20 \\
\hline
10 & 25 \\
\hline
\end{array}
\][/tex]
1. Understanding the Relationship:
- We observe that as the number of days overdue increases, the amount of fine increases. Checking the increments, we can see a pattern: for every 2 additional days overdue, the fine increases by 5 cents.
This suggests a linear relationship between the days overdue and the fine.
2. Identifying the Linear Relationship:
- A linear relationship can generally be expressed as [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Calculating the Slope (m):
- From the data:
- When days overdue [tex]\( x = 0 \)[/tex], fine [tex]\( y = 0 \)[/tex].
- When days overdue [tex]\( x = 10 \)[/tex], fine [tex]\( y = 25 \)[/tex].
- Using the formula for the slope, [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex]:
[tex]\[
m = \frac{25 - 0}{10 - 0} = \frac{25}{10} = 2.5
\][/tex]
4. Fitting the Equation:
- The linear equation, thus, can be represented as [tex]\( y = 2.5x \)[/tex].
5. Verifying the Model with All Data Points:
- Checking the other points using [tex]\( y = 2.5x \)[/tex]:
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 2.5 \times 2 = 5
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 2.5 \times 4 = 10
\][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[
y = 2.5 \times 6 = 15
\][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[
y = 2.5 \times 8 = 20
\][/tex]
- For [tex]\( x = 10 \)[/tex]:
[tex]\[
y = 2.5 \times 10 = 25
\][/tex]
- All given data points fit perfectly with [tex]\( y = 2.5x \)[/tex].
6. Deciding the Correct Equation:
- From the options provided:
- [tex]\( y = \frac{5}{2}x \)[/tex] is equivalent to [tex]\( y = 2.5x \)[/tex].
- We express [tex]\( y \)[/tex] (the fine) as a function of [tex]\( x \)[/tex] (the days overdue), hence:
[tex]\[
y = \frac{5}{2} x, \quad \text{where } y \text{ is the cost in cents for a book that is } x \text{ days overdue}.
\][/tex]
Therefore, the correct equation that best models the fine schedule for overdue books is:
[tex]\[
\boxed{y = \frac{5}{2} x, \text{ where } y \text{ is the cost in cents for a book that is } x \text{ days overdue.}}
\][/tex]
