Let's consider the problem step-by-step to understand the given linear relationship:
1. Understanding the Question:
We need to analyze the relationship between the number of miles traveled (x) and the total fee (y) and identify what the [tex]\( y \)[/tex]-intercept represents in this context.
2. Data Points:
The table provided gives the total fee for various distances:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of Miles} & \text{Total Fee} \\
\hline
2 & \$ 15.00 \\
\hline
5 & \$ 19.50 \\
\hline
7 & \$ 22.50 \\
\hline
10 & \$ 27.00 \\
\hline
15 & \$ 34.50 \\
\hline
\end{array}
\][/tex]
3. Concept of Linear Equation in Cab Fare:
The relationship between the distance (x) and the total fee (y) can be described using a linear equation of the form:
[tex]\[
y = mx + c
\][/tex]
where [tex]\( m \)[/tex] is the slope of the line and [tex]\( c \)[/tex] is the [tex]\( y \)[/tex]-intercept.
4. Slope (m):
The slope represents the rate of change in the fee with respect to distance. We determine it by taking any two points and computing:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the given data points, the calculations show that the slope (rate of increase per mile) is [tex]\( 1.5 \)[/tex].
5. Y-Intercept (c):
The [tex]\( y \)[/tex]-intercept in a linear equation represents the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. This is the starting fee or the base fare before any miles are traveled. The calculations indicate that this value (the base fare) is [tex]\( 12.0 \)[/tex].
6. Interpreting the Results:
Based on the results:
- The slope [tex]\( 1.5 \)[/tex] means that for every additional mile the cab travels, the total fee increases by \[tex]$1.50.
- The \( y \)-intercept \( 12.0 \) means that when the cab travels 0 miles, the total fee will be \$[/tex]12.00.
7. Final Answers:
- When the cab travels 0 miles, the total fee will be \[tex]$12.00.
- For every additional mile the cab travels, the total fee increases by \$[/tex]1.50.
Therefore, the correct interpretation for the [tex]\( y \)[/tex]-intercept in this context is:
When the cab travels 0 miles, the total fee will be \$ 12.00.
