Car Depreciation Problem: Calculating Future Value

Hey guys! Let's dive into a classic math problem involving car depreciation. This is the kind of question you might see on exams like UNICAMP, and it's super practical for understanding how things lose value over time. We'll break it down step-by-step, so you can tackle similar problems with confidence. So, let's get started and figure out how to calculate the future value of this car!

Understanding the Problem

Okay, so here’s the scenario: Two years ago, a certain car was valued at R$ 50,000.00, and now it's worth R$ 32,000.00. The question assumes that the car's value decreases at a constant annual rate. The goal is to figure out what the car will be worth in one more year. This involves understanding exponential decay, which is a key concept in mathematics and finance. Think of it this way: the car loses a percentage of its value each year, not a fixed amount. This is crucial because if it were a fixed amount, the calculation would be much simpler. We need to find this annual depreciation rate first. To make things clearer, we will explore the method of calculating car depreciation by understanding the annual rate.

• Initial Value: R$ 50,000.00
• Current Value: R$ 32,000.00
• Time Passed: 2 years
• Goal: Find the value in one more year.

The problem gives us enough information to set up an equation that models the car's depreciation. We know the starting value and the value after two years, which allows us to determine the annual depreciation rate. This rate is essential because it's the key to predicting the car's value in the future. Without understanding this rate, we're just guessing. So, our next step is to use this data to uncover that hidden percentage, which will then allow us to project the car's worth one year from now. This is where the math gets interesting, and we’ll see how to use formulas to solve real-world problems.

Calculating the Annual Depreciation Rate

To find the annual depreciation rate, we'll use the formula for exponential decay. This formula is your best friend when dealing with things that decrease in value over time, like cars, equipment, or even certain investments. The general form of the formula is:

Future Value = Initial Value * (1 - Depreciation Rate)^Number of Years

Let's break down each part:

• Future Value: This is the value of the car after a certain period (R$ 32,000.00 in our case).
• Initial Value: This is the car's original value (R$ 50,000.00).
• Depreciation Rate: This is what we're trying to find – the annual percentage decrease in value. Let's call it 'r'.
• Number of Years: This is the time that has passed (2 years).

Now, let's plug in the values we know into the formula:

32,000 = 50,000 * (1 - r)^2

Our next step is to solve for 'r'. First, we'll divide both sides of the equation by 50,000:

32,000 / 50,000 = (1 - r)^2

This simplifies to:

0.64 = (1 - r)^2

To get rid of the square, we take the square root of both sides:

√0.64 = 1 - r

Which gives us:

0.  8 = 1 - r

Now, we solve for 'r' by rearranging the equation:

r = 1 - 0.8
r = 0.2

So, the annual depreciation rate is 0.2, or 20%. This means the car loses 20% of its value each year. Now that we've found this crucial piece of information, we can move on to predicting the car's value in the next year. Remember, finding this rate is the key to solving the problem, so make sure you understand each step.

Predicting the Value in One Year

Now that we know the annual depreciation rate is 20%, we can predict the car's value in one year. We'll use the same exponential decay formula, but this time we're solving for the Future Value. The formula, as a reminder, is:

Future Value = Initial Value * (1 - Depreciation Rate)^Number of Years

This time, our 'Initial Value' is the current value of the car, which is R$ 32,000.00. The 'Depreciation Rate' is 20%, or 0.2, and the 'Number of Years' is 1, since we want to know the value in one year. So, let's plug in these values:

Future Value = 32,000 * (1 - 0.2)^1

First, we simplify the expression inside the parentheses:

Future Value = 32,000 * (0.8)^1

Since anything raised to the power of 1 is just itself, we have:

Future Value = 32,000 * 0.8

Now, we just multiply:

Future Value = 25,600

Therefore, the predicted value of the car in one year is R$ 25,600.00. This calculation shows how the car's value continues to decrease at the same annual rate. By understanding the formula and applying it correctly, we can make accurate predictions about future values in similar depreciation scenarios.

Analyzing the Answer and the Options

So, we've calculated that the car's value in one year will be R$ 25,600.00. Now, let's look at the multiple-choice options provided in the original problem. We have:

• (A) R$ 25,600.00
• (B) R$ 24,400.00
• (C) R$ 23...

Our calculated answer matches option (A). This confirms that our step-by-step process was correct. When you're taking a test, it's always a good idea to double-check your answer against the options provided. It can help you catch any simple mistakes or miscalculations. Also, noticing that our answer aligns with one of the options gives us extra confidence that we've solved the problem correctly. It's not just about getting the right number; it's about being sure of your method and understanding why your answer is the right one. So, in this case, we're not only sure of the answer, but we also understand the math behind it.

Key Takeaways and Tips for Similar Problems

Great job, guys! We’ve successfully solved this car depreciation problem. Let's recap the key steps and also discuss some tips that can help you handle similar questions in the future.

1. Understand the Concept of Exponential Decay: This is crucial. Recognize that the value decreases by a percentage each year, not a fixed amount.

2. Use the Correct Formula: Remember the formula for exponential decay:

   Future Value = Initial Value * (1 - Depreciation Rate)^Number of Years
   

   Make sure you know what each variable represents and how to plug in the values.

3. Calculate the Depreciation Rate: If the rate isn't given, you'll need to calculate it using the information provided, just like we did in this problem. This often involves solving for 'r' in the formula.

4. Predict Future Value: Once you have the depreciation rate, you can use it to predict the value at any point in the future.

5. Double-Check Your Work: Always verify your calculations and make sure your answer makes sense in the context of the problem. Also, check your answer against the multiple-choice options, if provided.

Tips for Similar Problems:

• Read the problem carefully: Make sure you understand what's being asked and what information is given.
• Identify the variables: Determine the Initial Value, Future Value, Depreciation Rate, and Number of Years.
• Show your work: This helps you track your steps and makes it easier to find mistakes.
• Practice, practice, practice: The more you solve these types of problems, the more comfortable you'll become with the process.

By following these steps and tips, you'll be well-equipped to tackle any car depreciation problem that comes your way. Remember, the key is to understand the underlying concepts and apply the correct formulas. Keep practicing, and you'll become a pro at these types of calculations!