Problem 89 - College Algebra with Modeling and Visualization

Question

The cost of driving a car includes both fixed costs and mileage costs. Assume that insurance and car payments cost \(\$ 350\) per month and gasoline, oil, and routine maintenance cost \(\$ 0.29\) per mile. (a) Find a linear function \(f\) that gives the annual cost of driving this car \(x\) miles. (b) What does the \(y\) -intercept on the graph of \(f\) represent?

Step-by-Step Solution

Answer

\( f(x) = 0.29x + 4200 \) and the \( y \)-intercept represents the annual fixed costs.

Step 1: Define Variables and Constants

We need to find a linear function, so let's define our variables and constants. The fixed costs per month are \(350, which include insurance and car payments. The variable cost, which is based on mileage, is \)0.29 per mile. Let's denote the number of miles driven annually as \( x \). Since there are 12 months in a year, the annual fixed cost will be \( 12 \times 350 \).

Step 2: Calculate Annual Fixed Cost

Calculate the annual fixed cost by multiplying monthly fixed cost by 12: \[ 350 \times 12 = 4200 \]. So, the annual fixed cost is \$4200.

Step 3: Establish Function Structure

A linear function for the total annual cost, \( f(x) \), includes the fixed and mileage costs. The fixed cost is constant \( 4200 \) and the mileage cost is \( 0.29x \). So, the function is: \[ f(x) = 0.29x + 4200 \].

Step 4: Understand the Y-intercept

In the linear function \( f(x) = 0.29x + 4200 \), the \( y \)-intercept, \( 4200 \), represents the total fixed costs of driving the car for a year, assuming no miles are driven. It accounts only for insurance and car payments, independent of mileage.

Key Concepts

Cost Analysis

Cost analysis involves breaking down the expenses associated with a specific activity or project. In the case of driving a car, we need to consider both fixed and variable costs to determine the overall expenditure. For a complete understanding, we differentiate the types of costs:

• Fixed costs stay the same regardless of how many miles you drive.
• Variable costs vary based on the distance driven and are usually calculated per mile.

By summing fixed and variable expenses, we can create a model that allows us to predict the total cost associated with driving different numbers of miles. This is crucial for budgeting and financial planning. It helps vehicle owners understand what they might expect to pay annually.

Variable Cost

Variable costs fluctuate depending on the level of activity or usage. In the context of car expenses, this is the cost linked to the actual driving.

• The given exercise highlights that gasoline, oil, and maintenance cost you \(0.29 \) per mile.
• The more miles you drive, the greater the variable costs accumulate.

To construct a linear function for total annual costs, these variable costs are represented as the slope, or "m," in the linear equation: \( f(x) = 0.29x + 4200 \).
Here, \(0.29x\) reflects the variable total based on the number of miles \(x\). Understanding this allows us to estimate how changes in driving habits can impact total expenses.

Fixed Cost

Fixed costs remain constant irrespective of the number of miles driven. In our car expense example, these are the monthly costs unrelated to mileage.

• The monthly fixed costs cover aspects like insurance and car payments.
• They remain at \(\$350 \) per month, giving an annual total when multiplied by 12 months.

This calculates to \( 12 \times 350 = 4200\). Hence, the fixed cost component of the function, \(4200\), does not change with the number of miles driven. It provides a baseline for costs that need to be accounted for regardless of vehicle usage.

Y-intercept

The y-intercept of a graph of a linear function is the point where the line crosses the y-axis. In our linear function equation for car costs, \( f(x) = 0.29x + 4200 \), the y-intercept is \(4200\).

• This value represents the total fixed costs of driving the car annually assuming no miles are driven.
• It's a reflection of expenses that do not vary with mileage, such as insurance and car payments.

In practical terms, should you decide not to drive at all for a year, you'd still incur \(\$4200 \) in fixed costs. This helps separate what part of your budget is immovable, which is vital for financial forecasting.