4.2 Modeling with linear functions  (Page 5/9)

It might help here to draw a picture of the situation. See [link] . It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates $\left(\text{3}0,\text{1}0\right),$ and Eastborough at $\left(\text{2}0,0\right).$

Using this point along with the origin, we can find the slope of the line from Westborough to Agritown.

Now we can write an equation to describe the road from Westborough to Agritown.

From this, we can determine the perpendicular road to Eastborough will have slope $m=–3.$ Because the town of Eastborough is at the point (20, 0), we can find the equation.

We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,

The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.

The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions in [link] .

A linear function is of the form $f(x)=mx+b.$ Using the rates of change and initial charges, we can write the equations

Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when $K(d)<M(d).$ The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the $K\left(d\right)$ function is smaller.

These graphs are sketched in [link] , with $K\left(d\right)$ in blue.

To find the intersection, we set the equations equal and solve:

This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that $K\left(d\right)$ is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is $d>100$ .