Problem 14 - Applied Calculus

Question

You are planning a trip whose principal cost is gasoline. (a) Make a table showing how the daily fuel cost varies as a function of the price of gasoline (in dollars per gallon) and the number of gallons you buy each day. (b) If your car goes 30 miles on each gallon of gasoline, make a table showing how your daily fuel cost varies as a function of your daily travel distance and the price of gas.

Step-by-Step Solution

Answer

Tables show fuel costs vary by gas price and distance traveled.

Step 1: Define Variables and Create a Structure for Table (a)

To construct the first table, consider the variables involved: the price of gasoline per gallon and the number of gallons purchased. Let the price of gasoline be denoted by \( p \) (dollars per gallon) and the number of gallons purchased daily be \( g \). The daily fuel cost is then calculated as \( \ ext{Fuel Cost} = p \times g \). The table will organize various values for \( p \) and \( g \) to show corresponding fuel costs.

Step 2: Populate Table (a) with Sample Calculations

Choose a range of values for \( p \) and \( g \). For instance:| Price per Gallon (\\() | Gallons Bought | Daily Fuel Cost (\\)) ||----------------------|----------------|------------------|| 2.50 | 10 | 25.00 || 3.00 | 10 | 30.00 || 3.50 | 10 | 35.00 || 4.00 | 12 | 48.00 || 4.50 | 12 | 54.00 |This table shows how varying prices and quantities affect daily expenses.

Step 3: Define Variables for Table (b)

For the second table, include the travel distance as an additional variable. Let \( d \) represent daily travel distance (miles) and \( m \) be the mileage (30 miles/gallon). The price of gas \( p \) remains the same. To find the number of gallons needed for a given distance, use \( g = \frac{d}{m} \). The daily fuel cost is \( \ ext{Fuel Cost} = p \times g = p \times \frac{d}{m} \).

Step 4: Populate Table (b) with Sample Calculations

Choose a set of travel distances and gas prices to illustrate different daily fuel costs:| Travel Distance (miles) | Price per Gallon (\\() | Daily Fuel Cost (\\)) ||-------------------------|----------------------|------------------|| 60 | 2.50 | 5.00 || 90 | 3.00 | 9.00 || 120 | 3.50 | 14.00 || 150 | 4.00 | 20.00 || 180 | 4.50 | 27.00 |This table evaluates how travel distance impacts fuel expenses given different prices.

Key Concepts

Price of Gasoline

Gasoline is a common fuel source needed for vehicles to function. The price of gasoline is crucial when it comes to planning travel expenses, as it is calculated per gallon. The cost of gasoline varies depending on several factors, including the region, the time of year, and market demand. A trip's fuel cost will increase if the price of gasoline rises. To calculate the daily fuel cost effectively, you need to multiply the amount you pay per gallon by the number of gallons you purchase. For example, if gasoline costs $3 per gallon and you buy 10 gallons, your total cost would be $30 for that day. Understanding how the price of gasoline impacts your budget can help you plan your trip more efficiently. It allows you to make conscious decisions and compare different costs to find saving opportunities.

Travel Distance

Travel distance is another important factor affecting fuel costs. It is the number of miles you intend to travel each day during a trip. Travel distance directly influences the amount of fuel needed, as longer distances require more fuel. To determine how much gas you'll need, you must know your car's mileage, which tells you how many miles per gallon your car can travel. For example, if you plan to cover 120 miles in a day and your car gets about 30 miles per gallon, you'll need \(\frac{120}{30} = 4\) gallons of gasoline. By calculating your travel distance and knowing your vehicle's mileage, you can estimate how much fuel you will need. In turn, this helps predict daily expenses based on distance and is essential for budgeting during travel.

Daily Expenses

Daily expenses while traveling can add up quickly, and fuel cost is a significant portion of these expenses. Understanding how to calculate your fuel costs accurately can help minimize the overall daily travel expenses. To figure out daily expenses, sum up all costs including fuel, lodging, meals, and other expenditures you might incur on a trip. Specifically, focus on the fuel cost by multiplying your daily travel distance in miles by the price per gallon, divided by the mileage: \(\text{Fuel Cost} = \left( \frac{\text{Travel Distance}}{\text{Mileage}} \right) \times \text{Price per Gallon} \). It's crucial to keep track of the daily expenses as it helps in staying within your budget and understanding where you can cut costs if necessary. Knowing daily fuel costs assists in making wise decisions on whether to adjust travel plans or find alternative solutions.

Mileage

Mileage refers to how many miles a vehicle can travel per gallon of gasoline. It’s often presented when you purchase or select a car and is important because it affects how much you'll spend on fuel during travels. For instance, a car with a mileage of 30 miles per gallon means that it consumes one gallon for every 30 miles traveled. Better mileage implies more efficient fuel consumption and less expenditure on gasoline. By knowing your vehicle's mileage, you can budget for road trips more accurately. This information helps you calculate how much gas you need based on your travel plans and estimate your daily fuel costs precisely. Optimizing mileage also involves maintaining the vehicle well and driving smoothly to ensure the vehicle performs at its best.