Problem 92 - College Algebra

Question

You will be developing functions that model given conditions. A car was purchased for \(\$ 22,500 .\) The value of the car decreases by \(\$ 3200\) per year for the first six years. Write a function that describes the value of the car, \(V\), after \(x\) years, where \(0 \leq x \leq 7 .\) Then find and interpret \(V(3)\)

Step-by-Step Solution

Answer

The value of the car after 3 years will be \$12900.

Step 1 : Formulate the function

Given the initial price of $22,500 and an annual depreciation amount of $3200, the value of the car after x years can be formulated as follows, V(x) = 22500 - 3200x.

Step 2 : Find the value of the car after 3 years

To find the value of the car after 3 years, substitute x = 3 into the formula. This results in V(3) = 22500 - 3200 * 3.

Step 3 : Calculate V(3)

Perform the calculation to get V(3) = 22500 - 9600 = \$12900.

Key Concepts

Linear Depreciation

Linear depreciation is a simple and straightforward way of calculating how an asset loses value over time. In this method, an asset depreciates by the same amount every year. This constant reduction makes linear depreciation easy to understand and apply.
For example, if a car is worth $22,500 initially and loses $3200 every year, this depreciation is linear.

• The rate of depreciation is constant at $3200 per year.
• This means the car's value goes down by $3200 each year, no matter what happens.

This predictable decrease allows us to model the value of the car with a simple linear function. That brings us to our next topic: modeling with functions.

Function Modeling

In algebra, function modeling means creating a mathematical function that represents a real-world situation.
In our example, we created a linear function to model the depreciation of a car.

• The function is formulated as: \( V(x) = 22500 - 3200x \).
• This equation shows how the car's value, \( V \), changes with each year, \( x \).

Here, \( 22500 \) is the initial value of the car, and \( 3200x \) signifies the yearly depreciation.
Using this function, you can easily calculate the car's value after any number of years within the range we have, which is 0 to 7 years in this situation.
Function modeling provides a powerful way to predict future values, offering clear insights into the process of depreciation.

Using Function Modeling in Real Life

Function modeling is not only useful in automotive depreciation but in other areas too, like electronics or machinery. Anywhere an asset decreases in value consistently over time, linear functions offer a neat solution.

Value Over Time

Understanding how value changes over time is essential for managing and predicting asset worth.
As in our car example, you see how depreciation impacts its value. By using our function \( V(x) = 22500 - 3200x \), you can assess the car's worth at any given year:

• For year 0 (the purchase year), the value is \( 22500 \).
• For year 3, calculated as \( V(3) = 22500 - 3200 \times 3 \).
• This results in a value of \( 12900 \).

Value over time helps track how money or investment evolves, useful in many financial planning scenarios.
In everyday usage, knowing the value over time is crucial for making informed decisions about when to sell or replace an asset.
It allows you to visualize the cost-effectiveness and longevity of different items, be it a car, a house, or any other investment worth keeping an eye on.