Chapter 1, Problem 20CT

To determine

To find : the linear equation for data and find the year in which the average monthly cost reached $\text{\$85 }$.

Answer to Problem 20CT

the linear equation is $C=2t+33.04$ , after $25.96$ years the average monthly cost reached $85$.

Explanation of Solution

Given information :

From the table provided in the question it can be observed that the

Cost for year $4$ is $41.04$.

Formula used :

The equation of line is $y=mx+c$.

Calculation: let $t=4$ represent $2004$ ,

Since $2004\text{ value }=\$41.04$ ,

So the point on linear equation is $\left(4,41.07\right)$.

From the table other point on linear equation is $\left(5,43.04\right)$

So for the rate of increasing of change is $\frac{43.04-41.04}{5-4}$

= $2$

So the equation is $C=2t+c$.

Now find the value of $c$ , substitute $t=4,C=41.04$

  $\begin{array}{l}C=2t+c\\ 41.04=2\left(4\right)+c\text{ [substitute t}=4,C=41.04]\\ c=33.04\text{ [simplify}]\end{array}$

Thus, the linear equation is $C=2t+33.04$.

For find the year substitute $C=85$ ,

  $\begin{array}{l}C=2t+33.04\\ 85=2t+33.04\text{ [substitute]}\\ 85-33.04=2t\text{ [subtract]}\\ 51.96=2t\text{ [subtract]}\\ t=25.96\text{ [divide by 2]}\end{array}$

Thus, after $25.96$ years the average monthly cost reached $85$.