The distance to the destination is given by a linear function of the total driving time. A general linear model can be written as:
[tex]D(t)=m\cdot t+b[/tex]
Where m and b are parameters. From the problem, we know that:
Driving time (minutes) Distance to destination (miles)
45 64.0
63 50.5
Using the model above and the table values, we construct the system of equations:
[tex]\begin{gathered} 64.0=45\cdot m+b \\ 50.5=63\cdot m+b \end{gathered}[/tex]
If we subtract these two equations:
[tex]\begin{gathered} 64.0-50.5=45\cdot m-63\cdot m+b-b \\ 13.5=-18\cdot m \\ m=-\frac{3}{4} \end{gathered}[/tex]
Now, we can use the first equation to calculate b:
[tex]\begin{gathered} 64.0=-45\cdot\frac{3}{4}+b \\ b=97.75 \end{gathered}[/tex]
Our linear model is:
[tex]D(t)=-0.75\cdot t+97.75[/tex]
For t = 71 minutes:
[tex]\begin{gathered} D(71)=-0.75\cdot71+97.75 \\ D(71)=44.5\text{ miles} \end{gathered}[/tex]
