Problem 4 - Algebra 1: Concepts and Skills

Question

You buy a used car for 7000 dollar. The car depreciates at the rate of 6% per year. Find the value of the car after the given number of years. $$5 years$$

Step-by-Step Solution

Answer

The value of the car after 5 years would be approximately $5572.67.

Step 1: Identify the initial value and the depreciation rate

In this problem, the initial value of the car is $7000 and the annual depreciation rate is 6%.

Step 2: Express the depreciation rate as a decimal

\(\)The depreciation rate 6% is expressed as 0.06 in decimal notation.

Step 3: Apply the depreciation formula.

We substitute the initial value, the depreciation rate and the number of years into this formula: `Initial value * (1 - depreciation rate) ^ number of years`. So, plug 7000 for the initial value, 0.06 for the depreciation rate and 5 for the years into the formula, we get \(7000 * (1 - 0.06)^5\)

Step 4: Calculate the future value of the car

Calculate the future value of the car by performing the above operation: \(7000 * (0.94)^5 = 5572.67\). So, the value of the car after 5 years would be approximately $5572.67

Key Concepts

Percentage Depreciation

Percentage depreciation is a measure of how much an asset decreases in value over a period. It is often expressed as a fixed percentage to denote the decrease in value annually. For instance, if a \(7,000 car depreciates at 6% per year, this means each year the car's value is reduced by 6% of its value at the beginning of the year. To calculate the value after a certain number of years, one needs to apply this rate repeatedly for the number of years in question.

In the case of the used car, with the depreciation rate given, we could calculate the depreciation amount for the first year as 6% of \)7,000, which is $420. However, for multiple years and for simplicity in calculation, we often use an algebraic expression to find the cumulative effect of the depreciation.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This concept is widely used in various scientific fields, including finance, to model depreciation. In our example, the value of the car after each year can be determined by multiplying the current value of the car by a factor that represents one year's worth of depreciation. Mathematically, this is expressed as an exponent in the depreciation formula: the base of \(1 - \text{depreciation rate}\) is raised to the power of the number of years.

Therefore, to find the value of the car after 5 years, we raise 0.94 (which is 1 minus the depreciation rate of 0.06) to the 5th power. This shows that the car's value doesn't just decrease by a fixed amount every year but that the decreased amount itself diminishes over time, which is a key characteristic of exponential decay.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They represent quantities in a generalized form and are powerful tools for modelling real-world scenarios. In the context of our problem, to represent the decreasing value of the car, we employ an algebraic expression to account for the changing value over time.

The formula \(7000 * (1 - 0.06)^5\) is an algebraic expression that succinctly models the car's value after 5 years considering a constant percentage depreciation rate. The structure of this formula is intentionally designed to embrace the concept of exponential decay, and as such, can be applied universally to any initial value and depreciation rate over any period.

Mathematical Finance

Mathematical finance is a field that uses mathematical models to solve problems related to finance, such as calculating depreciation, interest rates, and investment returns. The concepts of percentage depreciation and exponential decay are particularly critical in this domain because they help in understanding how assets lose value over time, which is essential for making informed financial decisions.

In practical terms, knowing how to calculate the future value of assets like a car after depreciation helps individuals and businesses in budgeting, tax preparation, and when making decisions about buying, selling, or replacing assets. The depreciation formula used in our car example is a basic yet indispensable tool in the toolkit of mathematical finance.