Question

A certain car rental location charges a daily fee as well as a mileage charge. On one trip a businessman rented a car for the day, drove it 150 miles. He was charged $79 (plus taxes). On another occasion when he rented a comparable car for the day, he drove it 65 miles and was charged $63.70 (plus taxes).

How many miles could the car be driven for $100 (without regard for any taxes that might be applied)? Round to a whole number of miles.

Answer 1

`x\approx 267\ miles`

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:

`f(x)=mx+b`

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:

`c = mx+b`

We must find the values of m and b with the data provided. Substituting the first point:

`79 = 150m+b`

Substituting the second point:

`63.70 = 65m+b`

Both equations form the following system:

`\left\{\begin{matrix}150m+b=79\\ 65m+b=63.70 \end{matrix}\right.`

Subtracting both equations:

`150m-65m=79-63.70`

Note the variable b was canceled out in the operation, leaving only the variable m to solve. Joining like terms:

`85m=15.3`

Solving:

`m=15.3/85=0.18`

From the first equation

`79 = 150m+b`

Solving for b:

`b=79-150m=79-150(0.18) = 52.`

The model for the problem is:

`c=0.18x+52`

Now we need to calculate how many miles (x) could be driven for c=$100. From the equation above, substitute c=100

`100=0.18x+52`

Solve for x:

`0.18x+52=100`

`0.18x=100-52=48`

`x=48/0.18=266.67`

Rounding to the closest integer:

`\boxed{x\approx 267\ miles}`