Answer:
[tex]x\approx 267\ miles[/tex]
Step-by-step explanation:
Linear Modeling
Some events can be modeled as linear functions. If we are in a situation where a linear model is suitable, then we need two sample points to make the model and predict unknown behaviors.
The linear function can be expressed in the slope-intercept format:
[tex]f(x)=mx+b[/tex]
For the problem at hand, we must pick the adequate variables according to the data provided.
The question states the charge for renting a car is a function of the mileage. It also provides two points from which we can build our model. Let's set the following variables:
c = the charge for renting a car in dollars
x = the distance driven by the businessman in miles
Representing the ordered pair as (x,c), we have the points: (150,79) and (65,63.70). Our model will be expressed as:
[tex]c = mx+b[/tex]
We must find the values of m and b with the data provided. Substituting the first point:
[tex]79 = 150m+b[/tex]
Substituting the second point:
[tex]63.70 = 65m+b[/tex]
Both equations form the following system:
[tex]\left\{\begin{matrix}150m+b=79\\ 65m+b=63.70 \end{matrix}\right.[/tex]
Subtracting both equations:
[tex]150m-65m=79-63.70[/tex]
Note the variable b was canceled out in the operation, leaving only the variable m to solve. Joining like terms:
[tex]85m=15.3[/tex]
Solving:
[tex]m=15.3/85=0.18[/tex]
From the first equation
[tex]79 = 150m+b[/tex]
Solving for b:
[tex]b=79-150m=79-150(0.18) = 52.[/tex]
The model for the problem is:
[tex]c=0.18x+52[/tex]
Now we need to calculate how many miles (x) could be driven for c=$100. From the equation above, substitute c=100
[tex]100=0.18x+52[/tex]
Solve for x:
[tex]0.18x+52=100[/tex]
[tex]0.18x=100-52=48[/tex]
[tex]x=48/0.18=266.67[/tex]
Rounding to the closest integer:
[tex]\boxed{x\approx 267\ miles}[/tex]
