Problem 106 - Precalculus Essentials

Question

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 function that represents the value of the car is \(V(x) = 22500 - 3200x \). After 3 years, the car value is \(V(3) = 22500 - 3200 * 3 = \$15700\). This means that three years after it was bought, the car is worth \$15700.

Step 1: Setting up the linear function

From the given problem, the value of the car decreases $3200 per year. This means each year, the value of the car is the previous year's value minus $3200. At time \(x=0\) (the start), the car is worth $22,500. So, a function to represent this can be written as \(V(x) = 22500 - 3200x \) where \(x\) is the number of years passed.

Step 2: Evaluating the function at \(x=3\)

To find \(V(3)\), substitute \(x = 3\) into the equation:\[ V(3) = 22500 - 3200*3 \]

Step 3: Interpreting the result

Evaluate \( V(3) \) to get the value of the car after three years. This value represents the monetary worth of the car three years after it was bought, taking into account the annual depreciation.

Key Concepts

Depreciation

Depreciation is an important financial concept that explains how the value of an asset, like a car, decreases over time. When you buy a car, it starts losing value the moment you drive it off the dealership. This loss is called depreciation. Depreciation can happen for several reasons:

• Normal wear and tear from driving and usage
• Obsolescence as newer models come with better features
• The general market trend and economic conditions

In the context of our car problem, we are considering straight-line depreciation, which means that the car loses the same amount of value every year. For our car, it depreciates by $3,200 each year. This is a constant rate of depreciation, making it a perfect case for using a linear function to model this loss over time.

Value of a Car

The value of a car can be considered its worth in monetary terms at any given point in time. When you purchase a car, its initial value is what you pay for it. However, as time passes, this value changes due to factors like depreciation. Imagine you bought a car for $22,500. According to the conditions laid out in our problem, this car decreases in value by $3,200 every year. So, after one year, the car would be worth $19,300, after two years $16,100, and so forth until after six years when it has decreased by this annual depreciation consistently. Understanding the value of a car at any time helps you make informed decisions about selling, buying, or trading in the vehicle. It is crucial to track this value to understand the impact of depreciation over the car's life.

Function Evaluation

Evaluating a function is a simple yet powerful mathematical tool. It allows us to find out exactly how the value of our car changes over time. We start by setting up a linear function using the initial car value and the annual depreciation rate. The function in this exercise is:\[V(x) = 22500 - 3200x\]Here, \(V(x)\) denotes the value of the car after \(x\) years. To evaluate this function, plug in any given value for \(x\), such as 3, to find the car's worth after that many years. Let's see how it works:1. Substitute \(x = 3\) into the function: \[ V(3) = 22500 - 3200(3) \]2. Simplify the equation: \[ V(3) = 22500 - 9600 \]3. Calculate the result: \[ V(3) = 12900 \]The function evaluation shows that after 3 years, the car's value is $12,900. This process helps in understanding real-life financial situations by using mathematics effectively.