Problem 57 - Precalculus: Mathematical for Calculus

Question

Depreciation The purchase value of an office computer is \(\$ 12,500 .\) Its annual depreciation is \(\$ 1875 .\) Find the value of the computer after 6 years.

Step-by-Step Solution

Answer

The computer's value after 6 years is $1,250.

Step 1: Identify Key Information

We start by identifying the given values in the problem. The initial value of the computer is \( \\(12,500 \). Each year the computer depreciates by \( \\)1875 \). We need to find the value after 6 years of depreciation.

Step 2: Calculate Total Depreciation Over 6 Years

Since the depreciation per year is \( \$1875 \), we need to multiply this by 6 years to find the total depreciation over that period.\[ \text{Total Depreciation} = 1875 \times 6 = 11250 \]

Step 3: Determine the Value After 6 Years

Now we subtract the total depreciation from the initial value of the computer to find its value after 6 years.\[ \text{Value after 6 years} = 12500 - 11250 = 1250 \]

Step 4: Verify the Result

Verify the calculations were done correctly, ensuring that the multiplication and subtraction were both completed accurately to find the final value.

Key Concepts

Annual Depreciation

When we talk about annual depreciation, we are referring to the decrease in value of an asset year by year. It's common for assets like computers, vehicles, and machinery to depreciate because they wear out or become outdated over time. Imagine you buy something new, and as each year passes, its value reduces a bit. That's what depreciation is all about.

In our exercise, the annual depreciation is given as $1875. This means the computer's value decreases by $1875 each year. Knowing this annual amount helps calculate how much value an asset loses over a specific period. This is essential when you need to track the asset's value for accounting or resale purposes.

Value Calculation

Value calculation involves determining the current worth of an asset after considering depreciation. To find the value of an asset after several years, you start with its original purchase value. Then, you subtract the total depreciation, which is the annual depreciation multiplied by the number of years.

In the given exercise, the computer's original value is $12,500. Over six years, it depreciates by a total amount of $11,250. Therefore, after these years:

• The total depreciation = $1875 (annual depreciation) × 6 years = $11,250.
• The current value = $12,500 (initial value) - $11,250 (total depreciation).

After these calculations, the remaining value of the computer is $1,250. This process helps in understanding how much an asset is worth at any given time, especially if you plan to sell or replace it.

Step-by-Step Solution

Breaking down problems into clear, manageable steps is key in solving depreciation calculations. This way, you avoid errors and ensure complete understanding.

Let's recap our step-by-step method for clarity: 1. **Identify Key Information**: Recognize the original value of the asset and its annual depreciation. For our computer, these are $12,500 and $1875 respectively. 2. **Calculate Total Depreciation**: Multiply the annual depreciation by the number of years to get the total depreciation. So, $1875 times 6 years equals $11,250. 3. **Determine Remaining Value**: Subtract the total depreciation from the original value. That calculation brings the computer's worth to $1,250 after 6 years. 4. **Verify the Result**: Always check your calculations. Ensuring each step is accurate helps prevent mistakes and gives confidence in your final answer.
Following these steps provides a reliable way to tackle depreciation problems and similar exercises. It's an approach that highlights the importance of organized and thoughtful resolution in mathematics.