Problem 40 - Algebra 2 and Trigonometry

Question

A car that cost \(\$ 20,000\) depreciated by 20\(\%\) each year. Find the value of the car at the end of each of the first four years. (A depreciation of 20\(\%\) means that the value of the car each year was 0.80 times the value the previous year.)

Step-by-Step Solution

Answer

The car's value at the end of the first four years is $16,000, $12,800, $10,240, and $8,192, respectively.

Step 1: Understand Initial Value and Depreciation Rate

The car initially costs $20,000. Each year, it depreciates by 20%, which means that each subsequent year's value is 80% (or 0.80 times) of the previous year's value.

Step 2: Calculate Value After First Year

To find the value after the first year, multiply the initial value by 0.80. \[ \text{Value after Year 1} = 20000 \times 0.80 = 16000 \]

Step 3: Calculate Value After Second Year

Use the end value from the first year to find the second year's value by multiplying again by 0.80. \[ \text{Value after Year 2} = 16000 \times 0.80 = 12800 \]

Step 4: Calculate Value After Third Year

Repeat the process to calculate the value after the third year using the end value from the second year. \[ \text{Value after Year 3} = 12800 \times 0.80 = 10240 \]

Step 5: Calculate Value After Fourth Year

Finally, calculate the car's value after the fourth year by multiplying the year three value by 0.80. \[ \text{Value after Year 4} = 10240 \times 0.80 = 8192 \]

Key Concepts

Depreciation Rate

Depreciation is a term that describes how the value of an asset decreases over time. This decrease in value occurs due to wear and tear, aging, or other factors like technological advancements. In our exercise, the car's price decreases by a certain percentage every year, which is known as the depreciation rate.

When we talk about a depreciation rate of 20%, it simply means that each year the car loses 20% of its value from the previous year. What's left after this decrease is the remaining 80% of the value. This 80% can be seen as multiplying the previous year's value by a factor of 0.80.

Understanding the depreciation rate is essential because it helps us forecast how much the asset, like a car, will be worth in the future.

• An initial value (e.g., car's cost)
• A percentage decrease every year (e.g., 20%)
• A multiplication factor (e.g., 0.80 for 80%)

With these, you can calculate the decreasing value over time.

Compound Interest

Although the exercise primarily focuses on depreciation, the concept of compound interest is related but works in the opposite way. Instead of losing value over time, compound interest refers to the process where an investment or amount of money grows over time.

In compound interest, the interest amount gets added to the principal, meaning the next interest calculation will be carried out on an increased total amount. This results in growth of both the principal and the interest earned from previous periods.

The formula for compound interest is typically: \[ A = P (1 + r)^n \]Where:

• \( A \) is the amount of money in the future
• \( P \) is the principal amount (initial investment)
• \( r \) is the annual interest rate
• \( n \) is the number of years

Similar to how assets depreciate by a percentage each year, compound interest adds a percentage annually, illustrating growth rather than decline.

Value After Years

To find out how much the car is worth after each year, you multiply the previous year's value by the remaining percentage after depreciation. In this case, since the car depreciates by 20%, its value after each year is 80% of what it was before.

For example, after the first year, you take the original cost of \(20,000 and multiply it by 0.80 to get \)16,000. For subsequent years, you continue this pattern using the result from the previous year. Here's a brief overview:

• Year 1: \( 20000 \times 0.80 = 16000 \)
• Year 2: \( 16000 \times 0.80 = 12800 \)
• Year 3: \( 12800 \times 0.80 = 10240 \)
• Year 4: \( 10240 \times 0.80 = 8192 \)

Each year you can see the value decrease, showing how depreciation accumulates over time. This pattern illustrates the exponential decay of the car’s value.