Problem 6 - Algebra for College Students

Question

If you pay \(\$ 12,000\) for a car and it depreciates \(20 \%\) per year, how much will it be worth in 5 years? Express your answer to the nearest dollar. \(\$ 3932\)

Step-by-Step Solution

Answer

The car will be worth $3932 after 5 years.

Step 1: Understand the Depreciation Formula

The formula for depreciation is given by \( V = P(1 - r)^t \), where \( V \) is the future value of the car, \( P \) is the initial price, \( r \) is the rate of depreciation, and \( t \) is the time in years.

Step 2: Identify Given Values

In this problem, \( P = 12,000 \) dollars, the annual depreciation rate \( r = 0.20 \), and the time \( t = 5 \) years.

Step 3: Substitute Values into the Formula

Substitute the values into the depreciation formula: \( V = 12,000(1 - 0.20)^5 \).

Step 4: Simplify the Expression Inside the Parentheses

Calculate \( 1 - 0.20 \) to get \( 0.80 \). Therefore, the expression becomes \( V = 12,000(0.80)^5 \).

Step 5: Calculate the Power of 0.80

Compute \( (0.80)^5 = 0.32768 \).

Step 6: Compute the Future Value

Multiply the initial price by this result: \( V = 12,000 \times 0.32768 = 3932.16 \).

Step 7: Round to the Nearest Dollar

Round \( 3932.16 \) to the nearest dollar, which is \( 3932 \).

Key Concepts

Depreciation

Depreciation is a method used to allocate the cost of a tangible asset over its useful life. This concept is important because many assets, like cars or machinery, lose their value over time.

• Definition: Depreciation represents how much of an asset’s value has been used up.
• Purpose: Understand an asset’s declining worth on financial statements.

In the exercise given, the car depreciates at a rate of 20% per year. An easy way to understand this is to think about the car losing a portion of its value annually. If an asset deprecates too quickly, it means you will have to replace it or lose money faster than expected. It's crucial when buying expensive items to understand how quickly they might depreciate.

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. This concept is used to describe situations where the rate of change is not constant but decreases over time, such as radioactive decay or depreciation of an asset.

• Formula: In mathematics, exponential decay can be represented by the formula: \( y = a(1 - r)^t \).
• Explanation: Here, \( y \) is the final amount, \( a \) is the initial amount, \( r \) is the decay rate, and \( t \) is time.

In our problem, a car's value over time is a classic example of exponential decay. Each year, the car loses 20% of its remaining value. By understanding exponential decay, we can predict that the car will lose its value rapidly after the purchase, with the most significant drop happening early on.

Future Value Calculation

Calculating the future value when exponential decay occurs requires understanding how values change over time with the given rate of decrease.

• Formula: This involves the use of \( V = P(1 - r)^t \), which helps in determining how much an asset will be worth in the future after accounting for depreciation.
• Application: In our example, \( V \) stands for the future value of the car, \( P \) is the initial cost (\\(12,000), \( r \) is the depreciation rate (0.20), and \( t \) is time (5 years).

The future value tells us what our asset will be worth after 5 years. With calculations, we found the car will be worth approximately \\)3,932. Understanding this calculation helps in planning finances, especially when buying vehicles or other depreciable assets.