Raj is deciding between two cell phone plans, [tex]$A$[/tex] and [tex]$B$[/tex], which are both linear functions. The monthly charge for plan [tex]$A$[/tex] according to the number of minutes used is shown in the table.

\begin{tabular}{|c|c|}
\hline
\multicolumn{2}{|c|}{ Monthly Charge for Plan A } \\
\hline
Minutes used, [tex]$x$[/tex] & Monthly charge (\[tex]$), $[/tex]y[tex]$ \\
\hline
0 & 14.45 \\
\hline
3 & 14.84 \\
\hline
6 & 15.23 \\
\hline
9 & 15.62 \\
\hline
12 & 16.01 \\
\hline
\end{tabular}

Plan $[/tex]B[tex]$ has the same monthly base charge as plan $[/tex]A[tex]$, but it charges a different amount per minute used. If the total monthly charge for plan $[/tex]B[tex]$ is $[/tex]\[tex]$22.10$[/tex] when 45 minutes are used, what is the slope of the linear function that represents the cost of plan [tex]$B$[/tex]?

A. 0.13
B. 0.17



Answer :

To solve this problem, we need to determine the slope of the linear function that represents the cost of Plan B.

### Step 1: Understand the structure of Plan A
Plan A's costs are given in the table, showing that the cost for Plan A is a linear function of minutes used. In a linear function of the form [tex]\( y = mx + b \)[/tex], [tex]\( m \)[/tex] is the slope (rate per minute) and [tex]\( b \)[/tex] is the y-intercept (base charge).

### Step 2: Calculate the slope ([tex]\(m\)[/tex]) and base charge ([tex]\(b\)[/tex]) of Plan A
The given points from the table for Plan A are (0, 14.45) and (3, 14.84).

- From (0, 14.45), we know that the y-intercept (base charge) [tex]\( b \)[/tex] is 14.45.
- The slope, [tex]\( m_A \)[/tex], can be calculated using the two points (0, 14.45) and (3, 14.84):

[tex]\[ m_A = \frac{(14.84 - 14.45)}{(3 - 0)} = \frac{0.39}{3} = 0.13 \][/tex]

Thus, the linear equation for Plan A is:
[tex]\[ y = 0.13x + 14.45 \][/tex]

### Step 3: Identify the base charge and total charge details of Plan B
Plan B has the same base charge ([tex]\( b \)[/tex]) as Plan A, which is 14.45. Given that the total monthly charge for Plan B with 45 minutes is 22.10, we can use this information to calculate the slope for Plan B.

### Step 4: Calculate the slope ([tex]\( m \)[/tex]) for Plan B
Using the information that the total charge for 45 minutes in Plan B is $22.10, we can set up the equation:

[tex]\[ 22.10 = m_B \cdot 45 + 14.45 \][/tex]

To solve for [tex]\( m_B \)[/tex]:

First, subtract the base charge from the total charge:
[tex]\[ 22.10 - 14.45 = 7.65 \][/tex]

Then, divide by the number of minutes (45):
[tex]\[ m_B = \frac{7.65}{45} = 0.17 \][/tex]

### Conclusion
The slope of the linear function representing the cost of Plan B is:
[tex]\[ \boxed{0.17} \][/tex]