Problem 24 - Algebra for College Students

Question

The Rent-Me Car Rental charges \(\$ 15\) per day plus \(\$ 0.22\) per mile to rent a car. Determine a linear function that can be used to calculate daily car rentals. Then use that function to determine the cost of renting a car for a day and driving 175 miles; 220 miles; 300 miles; 460 miles. See below

Step-by-Step Solution

Answer

The linear function is \( C(x) = 15 + 0.22x \). Costs: \( \$53.50 \), \( \$63.40 \), \( \$81.00 \), \( \$116.20 \).

Step 1: Define the Variables

Let's denote the total cost of renting the car for a day and driving a certain number of miles as \( C \). Let \( x \) represent the number of miles driven. The task is to express \( C \) as a function of \( x \).

Step 2: Identify the Fixed and Variable Costs

The Rent-Me Car Rental charges a fixed cost of \( \\(15 \) per day (regardless of how many miles are driven), plus an additional charge of \( \\)0.22 \) for every mile driven.

Step 3: Formulate the Linear Function

Given the fixed cost for a day and the per-mile charge, the total cost \( C \) can be described by the linear function: \[ C(x) = 15 + 0.22x \]. This function sums up the fixed daily rate and the product of miles driven and the per-mile rate.

Step 4: Calculate the Cost for 175 Miles

Substitute \( x = 175 \) into the function: \[ C(175) = 15 + 0.22 \times 175 = 15 + 38.5 = 53.5 \]. Thus, the cost for driving 175 miles is \( \$53.50 \).

Step 5: Calculate the Cost for 220 Miles

Substitute \( x = 220 \) into the function: \[ C(220) = 15 + 0.22 \times 220 = 15 + 48.4 = 63.4 \]. Thus, the cost for driving 220 miles is \( \$63.40 \).

Step 6: Calculate the Cost for 300 Miles

Substitute \( x = 300 \) into the function: \[ C(300) = 15 + 0.22 \times 300 = 15 + 66 = 81 \]. Therefore, the cost for driving 300 miles is \( \$81.00 \).

Step 7: Calculate the Cost for 460 Miles

Substitute \( x = 460 \) into the function: \[ C(460) = 15 + 0.22 \times 460 = 15 + 101.2 = 116.2 \]. Hence, the cost for driving 460 miles is \( \$116.20 \).

Key Concepts

Variable Costs

Variable costs change depending on how many units of something are used or produced. In car rentals, these are the costs that increase with each mile you drive.
For example, if you rent a car from Rent-Me Car Rental, one of the variable costs is the per-mile fee of $0.22.
This cost will add up as you cover more distance with the car.

• The more miles you drive, the higher your variable cost.
• If you drive 0 miles, your variable cost is $0 because it depends entirely on mileage.

Understanding variable costs is important because they help you figure out how much your trip will cost, based on the miles you'll be driving.

Fixed Costs

Fixed costs remain the same no matter how many miles you drive. In the context of a car rental, this is the standard daily fee you pay just to have access to the car.
For Rent-Me Car Rental, the fixed cost is $15 per day.

• This $15 charge applies whether you drive the car or not.
• Fixed costs help you predict expenses because they don't change.

Knowing your fixed costs helps you budget for the base cost of renting, regardless of your travel plans.

Cost Calculation

Cost calculation is the process of figuring out the total cost of a transaction or service, such as renting a car.
To calculate the total cost of a car rental, you add fixed costs and variable costs.
For Rent-Me Car Rental, the cost calculation formula is a linear function:\[ C(x) = 15 + 0.22x \]Here, \( C(x) \) represents the total cost, which is determined by adding the fixed cost \( 15 \) dollars and the variable cost \( 0.22x \), where \( x \) is the number of miles driven.

• This formula lets you easily compute the total rental cost for any given mileage.
• Just plug in the number of miles you plan to drive to get the total cost.

Effective cost calculation lets you know in advance how much to expect to pay for using the car.

Mile Rate

The mile rate is the cost you incur per mile when you drive a rental car. Essentially, it's the price tag attached to each mile on your trip.
At Rent-Me Car Rental, the mile rate is $0.22.

• This rate is key for understanding how variable costs will increase as you drive more miles.
• With a clear mile rate, you can easily predict the additional costs for traveling further.

Understanding the mile rate allows you to plan your trips better and avoid unexpected expenses from driving extra miles.