Sure, let's analyze the given data and find out the cost structure of the wireless company's international texting plan.
The data provided is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of text messages sent } (x) & \text{Total cost } (y) \\
\hline
10 & \$6.50 \\
\hline
15 & \$6.75 \\
\hline
20 & \$7.00 \\
\hline
25 & \$7.25 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution
1. Understanding the Problem:
- The total cost [tex]\( y \)[/tex] is a linear function of the number of text messages sent [tex]\( x \)[/tex].
- The general form of the linear equation is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\( m \)[/tex] is the slope of the line (the additional cost per text message) and [tex]\( b \)[/tex] is the y-intercept (the flat access fee).
2. Linear Regression Analysis:
- We need to determine the values of [tex]\( m \)[/tex] (additional cost per text message) and [tex]\( b \)[/tex] (access fee).
3. Data Points:
- We have four data points: [tex]\((10, 6.50)\)[/tex], [tex]\((15, 6.75)\)[/tex], [tex]\((20, 7.00)\)[/tex], and [tex]\((25, 7.25)\)[/tex].
4. Finding the Slope (m):
- For simplicity, let's denote the slope [tex]\( m \)[/tex] as the change in the total cost ([tex]\( y \)[/tex]) divided by the change in the number of texts ([tex]\( x \)[/tex]):
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- From the data points provided, it can be observed that as the number of text messages increases by 5, the total cost increases by 0.25 incrementally. This consistent increment gives us enough basis to understand the slope.
5. Finding the Intercept (b):
- Once we have the slope [tex]\( m \)[/tex], we can use any of the data points to solve for [tex]\( b \)[/tex]. For example, using the point [tex]\((10, 6.50)\)[/tex]:
[tex]\[
6.50 = m \cdot 10 + b
\][/tex]
6. Substituting the Values:
- Through calculations, it was found that the additional cost per text message [tex]\( m \)[/tex] is 0.05.
- Substituting [tex]\( m = 0.05 \)[/tex] in the equation [tex]\( y = 0.05x + b \)[/tex] and solving with any point, we find out the access fee [tex]\( b \)[/tex].
Therefore:
- The additional cost per text message is [tex]\( \boxed{0.05} \)[/tex].
- The flat access fee is [tex]\( \boxed{6.00} \)[/tex].
To summarize:
The wireless company charges a flat access fee of [tex]\( \$6.00 \)[/tex] and an additional [tex]\( \$0.05 \)[/tex] per text message sent.
