Problem 106 - College Algebra Essentials

Question

In Exercises \(105-108,\) you will be developing functions that model given conditions. A car was purchased for \(\$ 22,500\). The value of the car decreased by \(\$ 3200\) per year for the first six years. Write a function that describes the value of the car, \(V,\) after \(x\) years, where \(0 \leq x \leq 6 .\) Then find and interpret \(V(3)\)

Step-by-Step Solution

Answer

The car would be worth \$12,900 after three years.

Step 1: Identify Value Components

Identify the initial value of the car and the rate of depreciation. In this scenario, the initial value is \(\$22,500\) and the depreciation rate is \(\$3,200\) per year.

Step 2: Develop the Function

Develop a function that represents the car's value over time. Considering linear depreciation, the value of the car after \(x\) years, \(V(x)\), is given by the initial value minus the product of the depreciation rate and time, that is \(V(x) = \$22,500 - \$3,200x\). This function holds for \(0 \leq x \leq 6\).

Step 3: Evaluate the Function

Evaluate the function for \(x = 3\) to obtain the value of the car after three years. Substitute \(x = 3\) into the function: \(V(3) = \$22,500 - \$3,200 * 3\).

Step 4: Compute the Car Value

Perform the computation to find the car's value after three years: \(V(3) = \$22,500 - \$9,600 = \$12,900\).

Key Concepts

Value Function

In understanding linear depreciation, it's crucial to comprehend the concept of a value function. A value function, often denoted as \( V(x) \), is a mathematical expression that represents how the value of something, like a car, changes over time. When it comes to financial assessments and estimations, this function helps us track the loss or decrease in value over specified periods. For objects like cars, the "V" typically represents the value at time "x" years. Linear depreciation implies that the value decreases by a consistent amount each year. The formula for a linear value function is generally structured as:

• \( V(x) = \text{Initial Value} - (\text{Depreciation Rate} \times x) \)

This formula provides an easy pathway to understanding how the value changes annually.

Initial Value

The initial value is where we begin our examination of depreciation. It is essentially the starting point or the price for which an asset is purchased. In this exercise, the car's initial value is given as $22,500. This amount is critical as it represents the car's monetary worth at \( x = 0 \) years, which marks the beginning of our evaluation timeline. Understanding the initial value helps set the context for further calculations. It's the foundation from which depreciation occurs and is a fixed figure used in the value function.

Depreciation Rate

A core component of evaluating a value function involves understanding the depreciation rate. This refers to how much the value of an asset declines over a specified time frame, such as annually. In the exercise above, the depreciation rate is $3,200 per year.It’s a linear rate, meaning this same amount is subtracted from the car’s value each year. This concept helps in predicting future values and financial planning. The constant rate simplifies calculation, as shown in the formula:

• \( ext{Depreciation over } x ext{ years=} \text{Depreciation Rate} \times x \)

This regular decrement reflects how quickly the asset loses worth over time.

Evaluation of Function

Solving for a specific point in time using the value function is referred to as evaluation. Essentially, it involves substituting a specific value of "x" (time in years) into the function to derive the asset’s value at that point.For instance, when evaluating the function for \( x = 3 \), you replace "x" in the equation:

• \( V(3) = \\(22,500 - \\)3,200 \times 3 \)
• \( V(3) = \\(22,500 - \\)9,600 \)
• \( V(3) = \\(12,900 \)

This calculated value of \)12,900 represents the worth of the car three years after purchase, illustrating how evaluations provide detailed financial insights about asset depreciation over time.