Problem 40 - Calculus and its applications

Question

At the beginning of a trip, the odometer on a car reads \(30,680,\) and the car has a full tank of gas. At the end of the trip, the odometer reads \(31,077 .\) It takes 13.5 gal of gas to refill the tank. a) What is the average rate at which the car was traveling, in miles per gallon? b) What is the average rate of gas consumption in gallons per mile?

Step-by-Step Solution

Answer

a) 29.41 mpg b) 0.034 gallons per mile

Step 1: Determine Total Miles Traveled

Subtract the initial odometer reading from the final odometer reading to find the total distance traveled during the trip. \[\text{Total Miles Traveled} = 31,077 - 30,680 = 397 \text{ miles}\]

Step 2: Calculate Average Miles Per Gallon

To find the average miles per gallon, divide the total miles traveled by the amount of gas used to refill the tank.\[\text{Miles Per Gallon} = \frac{397 \text{ miles}}{13.5 \text{ gallons}} \approx 29.41 \text{ mpg}\]

Step 3: Calculate Average Gallons Per Mile

To find the average gallons per mile, take the reciprocal of the number of miles per gallon.\[\text{Gallons Per Mile} = \frac{1}{29.41} \approx 0.034 \text{ gallons per mile}\]

Key Concepts

Miles Per Gallon

When driving, we often want to know how efficiently our car uses fuel. "Miles per gallon," or mpg, is a common measure of this efficiency. It tells us how many miles a car can travel using one gallon of gasoline. Calculating mpg is useful for understanding your vehicle's fuel economy.

To find the miles per gallon, you'll first need to know two things:

• The total distance traveled.
• The total amount of gasoline used.

For example, if your car’s odometer read 30,680 miles at the start of your trip and 31,077 miles at the end, the total distance traveled is the difference of these values: 397 miles.

If you filled up the tank with 13.5 gallons of gasoline at the end, you use these amounts to calculate mpg. Divide the total miles (397) by the gallons used (13.5):\[\text{Miles Per Gallon} = \frac{397}{13.5} \approx 29.41 \text{ mpg}\] This calculation shows you the car can travel approximately 29.41 miles on one gallon of gas. It's a handy number for comparing the efficiency of different vehicles or understanding your own driving habits.

Gallons Per Mile

"Gallons per mile" is another way to evaluate fuel efficiency, telling us how much fuel is used to travel one mile. It's essentially the reciprocal of miles per gallon, making it another useful measure of a car's fuel usage.

To compute gallons per mile, take the mpg value you already calculated and find its reciprocal:\[\text{Gallons Per Mile} = \frac{1}{\text{Miles Per Gallon}}\]In our example, if the car's efficiency is 29.41 miles per gallon, then the average gallons per mile would be:\[\text{Gallons Per Mile} = \frac{1}{29.41} \approx 0.034 \text{ gallons per mile}\] This tells us the car uses about 0.034 gallons to go one mile. This metric can be helpful in cost calculations, as it directly relates gas consumption to distance.

Distance Traveled Calculation

Calculating the distance traveled during a trip is straightforward when you have an odometer. The odometer measures the total miles driven by the vehicle.

For any journey, you can determine the miles traveled by taking two odometer readings:

• The reading at the start of your trip.
• The reading at the end of your trip.

Subtract the starting value from the ending value to find the total miles traveled.

For instance, with a starting odometer reading of 30,680 miles and an ending reading of 31,077 miles, the calculation is:\[\text{Total Miles Traveled} = 31,077 - 30,680 = 397 \text{ miles}\]This step is crucial because knowing the exact distance traveled allows for accurate fuel efficiency evaluations, turning raw data into useful information for drivers wanting to optimize their fuel usage.