The first charges an up front fee of $20 then 59 cents a mile the second charges an up front fee of $16 then 63 cents a mile when will the first be better than the second

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.

Input

d, distance driven in miles

Outputs

K(d): cost, in dollars, for renting from Keep on TruckingM(d) cost, in dollars, for renting from Move It Your Way

Initial Value

Up-front fee: K(0) = 20 and M(0) = 16

Rate of Change

K(d) = $0.59/mile and P(d) = $0.63/mile

A linear function is of the form  

f(x)=mx+b\displaystyle f\left(x\right)=mx+b

f(x)=mx+b. Using the rates of change and initial charges, we can write the equations

{K(d)=0.59d+20M(d)=0.63d+16\displaystyle \begin{cases}K\left(d\right)=0.59d+20\\ M\left(d\right)=0.63d+16\end{cases}

{

​K(d)=0.59d+20

​M(d)=0.63d+16

​​

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(d)\displaystyle K\left(d\right)

K(d) function is smaller.

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(d) 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\displaystyle d>100

d>100.

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