Question

the monthly cost (in dollars) of a long distance phone plans in a linear function of the total calling time(in minutes). the monthly cost for 32 minutes of calls is $15.87 and the monthly cost for 54 minutes is $17.85 what is the monthly cost for 34 minutes of calls?

Answer 1

We will use a linear function model to express a relation between the charges and the time for calls.

The linear model is expressed in a general form:

$C\text{ = m}\cdot t\text{ + b}$

The above form resembles the linear ( slope-intercept ) relaitonship. Where,

$\begin{gathered} m\text{ = rate charged by network company} \\ b\text{ = base charges} \end{gathered}$

We see that there are two parameters to highlight the linear calling charges model. We will determine each one by one and define the model.

The rate charged by the network company can be evaluated from the given set of pairs of call charges and the time taken for the call. We will go ahead and outline them below:

$\begin{gathered} (\text{ charges , time )} \\ (\text{ \ 15.87 , 32 mins )} \\ (\text{ \ 17.85 , 54 mins ) } \end{gathered}$

The parameter ( m ) can be evaluated as follows:

$m\text{ = }\frac{charge_2-charge_1}{time_2-time_1}$

Evaluate ( m ):

$\begin{gathered} m\text{ = }\frac{17.85\text{ - 15.87}}{54\text{ - 32}} \\ \\ m\text{ = }\frac{1.98}{22} \\ \\ m\text{ = }0.09 \end{gathered}$

To determine the parameter ( b ) we will use any one set of data and the modify our linear model as follows:

$\begin{gathered} \text{Modified Model:} \\ C\text{ = 0.09}\cdot t\text{ + b} \end{gathered}$

Data:

$(\text{ \ 15.87 , 32 mins)}$

We will plug in the data point in the modified equation to evaluate for ( b ):

$\begin{gathered} 15.87\text{ = 0.09}\cdot(32)\text{ + b } \\ b\text{ = }12.99 \end{gathered}$

We can complete our linear model by plugging in the values of the parameters:

$\begin{gathered} \text{Complete Model:} \\ C\text{ = 0.09}\cdot t+12.99 \end{gathered}$

We will use the model above to determine the cahrges for 34 mins of call. Therefore, plugging in t = 34:

$\begin{gathered} C\text{ = 0.09}\cdot(34)\text{ + 1}2.99 \\ C\text{ = 3.06 + }12.99 \\ C\text{ = \ }16.05 \end{gathered}$

Therefore, the solution is:

$16.05$