Question

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 forgot to refuel the vehicle when he returned it. his check - out sheet indicates that the vehicle had about 1/2 of a tank left. the tanks is now nearly empty.
if it takes 11.75 gallons of gas to refuel the car tim rented, how much extra should he expect to pay because he forgot to get gas?
a. $26.44
b. $31.44
c. $52.88
d. $57.88

Explanation:

Step1: Determine the amount of gas needed

The vehicle had \( \frac{1}{2} \) of a tank left, and it takes 11.75 gallons to refuel. So the amount of gas needed is \( 11.75\times\frac{1}{2}= 5.875 \) gallons? Wait, no. Wait, if the tank was half full, and to fill it completely is 11.75 gallons? Wait, no, the problem says "it takes 11.75 gallons of gas to refuel the car Tim rented" when he forgot. Wait, the Check - Out sheet indicates the vehicle had about \( \frac{1}{2} \) of a tank left. So the amount of gas used (needed to refuel) is the total capacity (11.75 gallons) minus the gas left. Since it had \( \frac{1}{2} \) tank left, the gas left is \( 11.75\times\frac{1}{2}=5.875 \) gallons. So the gas needed to refuel is \( 11.75 - 5.875=5.875 \) gallons? Wait, no, maybe the tank capacity is 11.75 gallons? Wait, the problem says "it takes 11.75 gallons of gas to refuel the car Tim rented" because he forgot to get gas. Wait, maybe the tank was half empty? Wait, let's re - read: "Tim forgot to refuel the vehicle when he returned it. His Check - Out sheet indicates the vehicle had about \( \frac{1}{2} \) of a tank left. The tanks is now nearly empty. If it takes 11.75 gallons of gas to refuel the car Tim rented, how much extra should he expect to pay because he forgot to get gas?"

Ah, so the tank capacity is such that to fill it from nearly empty (after he returned it) to full is 11.75 gallons? Wait, no. Wait, when he rented it, the tank had \( \frac{1}{2} \) tank. When he returned it, the tank is nearly empty. So the amount of gas he used is \( \frac{1}{2} \) of the tank's capacity. And the tank's capacity is 11.75 gallons? Wait, no, the cost is $4.50 per gallon plus a $5.00 refueling charge.

Wait, let's parse the cost: "Renter's total charge at a cost of $4.50 per gallon plus a $5.00 refueling charge."

First, find the number of gallons needed: The vehicle had \( \frac{1}{2} \) tank left, and to refuel (from nearly empty) to full is 11.75 gallons? Wait, no. The Check - Out sheet says the vehicle had about \( \frac{1}{2} \) of a tank left. So the amount of gas that should have been in the tank (at checkout) was \( \frac{1}{2} \) of the tank's capacity. When he returned it, the tank is nearly empty, so the gas he used is \( \frac{1}{2} \) of the tank's capacity. And the tank's capacity is such that refueling (from empty) to full is 11.75 gallons? Wait, maybe the tank's capacity is 11.75 gallons. So when he rented it, the tank had \( \frac{1}{2}\times11.75 = 5.875 \) gallons. When he returned it, the tank is nearly empty (0 gallons). So the gas he used (and needs to pay for) is \( 5.875 \) gallons? Wait, no, the problem says "it takes 11.75 gallons of gas to refuel the car Tim rented" because he forgot to get gas. So the amount of gas to refuel is 11.75 gallons? Wait, maybe I misread. Let's start over.

The cost per gallon is $4.50, plus a $5.00 refueling charge.

The amount of gas needed: The vehicle had \( \frac{1}{2} \) tank left at checkout. When he returned it, the tank is nearly empty. So the gas he used is \( \frac{1}{2} \) of the tank's capacity. And the tank's capacity is 11.75 gallons (since it takes 11.75 gallons to refuel). So the gas needed is \( 11.75\times\frac{1}{2}=5.875 \) gallons? Wait, no, if the tank was half full at checkout, and he returned it empty, then the gas he used is half the tank. And the tank's capacity is 11.75 gallons, so the gas used is \( 11.75\times\frac{1}{2} = 5.875 \) gallons.

Now, calculate the cost: Cost = (gallons × cost per gallon) + refueling charge.

Gallons = 5.875, cost per gallon = $4.50, refueling charge = $5.00.

Step2: Calculate the cost

First, calculate the cost of the gas: \( 5.875\times4.50=\frac{47}{8}\times\frac{9}{2}=\frac{423}{16}=26.4375\approx26.44 \)? Wait, no, that can't be. Wait, maybe the tank's capacity is 11.75 gallons, and he used all of the half - tank. Wait, maybe I got the gas amount wrong. Wait, the problem says "it takes 11.75 gallons of gas to refuel the car Tim rented" because he forgot to get gas. So the amount of gas to refuel is 11.75 gallons? Wait, no, the Check - Out sheet says the vehicle had about \( \frac{1}{2} \) of a tank left. So the gas that should have been in the tank (at checkout) was \( \frac{1}{2} \) of the tank's capacity. When he returned it, the tank is nearly empty, so the gas he used is \( \frac{1}{2} \) of the tank's capacity. And the tank's capacity is 11.75 gallons, so the gas used is \( 11.75\times\frac{1}{2}=5.875 \) gallons.

Now, calculate the cost:

Cost = (number of gallons × cost per gallon) + refueling charge

Number of gallons = 5.875, cost per gallon = $4.50, refueling charge = $5.00

Cost of gas = \( 5.875\times4.50=\frac{47}{8}\times\frac{9}{2}=\frac{423}{16}=26.4375\approx26.44 \)

Then add the refueling charge: \( 26.44 + 5.00=31.44 \)? Wait, no, the refueling charge is a flat $5? Wait, the problem says "Renter's total charge at a cost of $4.50 per gallon plus a $5.00 refueling charge." So total cost = (gallons × 4.50)+5.00

Wait, if the amount of gas needed is 5.875 gallons (since the tank was half full, and to fill it is 11.75 gallons, so half is 5.875), then:

Gallons = 5.875

Cost of gas: \( 5.875\times4.50 = 26.4375\approx26.44 \)

Refueling charge: $5.00

Total cost: \( 26.44+5.00 = 31.44 \)

Wait, that's option b.

Wait, let's check the calculation again:

11.75 gallons is the amount to refuel (from empty to full). The vehicle had \( \frac{1}{2} \) tank left, so the gas used (the gas that needs to be refueled) is \( 11.75\times\frac{1}{2}=5.875 \) gallons.

Cost per gallon: $4.50, so cost of gas: \( 5.875\times4.50 = 26.4375\approx26.44 \)

Refueling charge: $5.00

Total cost: \( 26.44 + 5.00=31.44 \)

Yes, that's $31.44, which is option b.

Answer:

b. $31.44