Question

The following section is a statement from the rental agreement Tim signed when he rented his car this past weekend. "Upon checkout, the fuel level of the vehicle will be determined by turning the vehicle on and visually inspecting the fuel gauge. The approximate fuel level will be recorded on the Check-Out sheet and verified with initials by the vehicle Renter. One copy of the Check-Out sheet will be given to the customer. Another copy will be kept with the on-site records of the vehicle. "The rented vehicle must be returned with a minimum fuel level the same as that indicated on the Check-Out sheet. A vehicle returned with a fuel level less than the approximate level indicated on the Check-Out sheet will be completely refueled with on-site pumps. The price of the fuel used to refuel the vehicle will be added to the Renter's total charge at a cost of $4.50 per gallon plus a $5.00 refueling charge." Tim is running extremely late and has just 5 minutes to return his rental car to the agency. The fuel tank in the car is half empty. It was full when he picked it up. (The car has a 12-gallon fuel tank.) He doesn't think he can get gas and get to the agency to return the car in time. Returning the car a day late will result in an extra daily rental charge of $32.00 and refueling the car on his own will cost $2.59 per gallon at the local gas station. Which of the following statements is true? a. It would be cheaper to refuel the car and return it a day late. b. It would be cheaper to return the car on time and pay the refueling costs. c. It would cost about the same to return the car on time without gas or one day late with gas. d. Tim is not responsible for refueling the car when he returns it.

Answer 1

To check how much Tim will pay to refill the gas tank using one of the on-site pumps, we can create a linear model. We know from our problem that the Renter's charge $4.50 per gallon plus a $5.00 refueling, so let $x$ be the number of gallons and $y$ the total cost.
$y=4.50x+5$ 
We also know that the fuel tank in the car is half empty and the car has a 12-gallon fuel tank, so half tank is 6 gallons. Lets replace the number of gallons in our linear model with 6 to fin the total cost: 
$y=4.50x+5$ 
$y=4.50(6)+5$
$y=27+5$
$y=32$

Now, to find how much Tim will pay if he decides to refuel the car on his own, we can also create a linear model.
Let $x$ be the number of gallons and $y$ the total cost. We know form our problem that refueling the car on his own will cost $2.59 per gallon and the late fee charge is $32.
$y=2.59x+32$
We already know that half tank is 6 gallons, so 
$y=2.59(6)+32$
$y=15.54+32$
$y=47.54$

Comparing the result of our two linear models we can conclude that the correct answer is: b. It would be cheaper to return the car on time and pay the refueling costs.