Which Expression Can Be Used To Determine The Value Of The Car After T Years, Given That Sunny Purchased A New Car For $29,873 And It Depreciates 20% Annually?
Let's delve into the fascinating world of depreciation by examining a real-world scenario. Sunny has made a significant investment by purchasing a brand-new car for $29,873. However, like most assets, cars lose value over time – a concept known as depreciation. In Sunny's case, the car depreciates at a rate of 20% annually. Our mission is to determine which expression accurately represents the car's value after t years. This involves understanding the principles of exponential decay and applying them to a practical situation.
When dealing with depreciation, we're essentially working with a decreasing value over time. The initial value of the asset (in this case, Sunny's car) serves as our starting point. The depreciation rate, expressed as a percentage, tells us how much the value decreases each year. To calculate the value after a certain number of years, we need to use a formula that accounts for this annual reduction. The formula for exponential decay is a perfect fit for this scenario. It allows us to model the decreasing value of the car as time passes. Let's break down the key components of this formula and see how they apply to Sunny's car.
The core concept here is that the car retains a certain percentage of its value each year. If the car depreciates by 20%, it means that it retains 80% of its value. This retained percentage is crucial for calculating the value after multiple years. We'll express this percentage as a decimal (0.80 in this case) and use it as the base of our exponential function. The exponent, t, represents the number of years that have passed. As t increases, the value of the car decreases, reflecting the depreciation. Understanding this relationship between depreciation rate, retained value, and time is fundamental to solving this problem. We'll explore how different expressions capture this relationship and identify the one that accurately models Sunny's car depreciation.
Depreciation plays a crucial role in understanding the value of assets over time. This is especially important for items like cars, machinery, and equipment, which naturally lose value as they age and are used. The concept of exponential decay comes into play when the value decreases at a constant percentage rate over a period. In our scenario, Sunny's car depreciates at a consistent 20% each year. This consistent percentage decrease is a hallmark of exponential decay, making it the perfect mathematical model to represent the car's value over time.
The exponential decay formula is generally expressed as:
Value after t years = Initial Value * (1 - Depreciation Rate)^t
Let's break down this formula to understand each component's role. The Initial Value is the starting value of the asset, which in Sunny's case is the purchase price of $29,873. The Depreciation Rate is the percentage by which the value decreases each year, given as 20% or 0.20 in decimal form. The term (1 - Depreciation Rate) represents the percentage of value retained each year. For Sunny's car, this is (1 - 0.20) = 0.80, meaning the car retains 80% of its value annually. The exponent t signifies the number of years that have passed. The formula essentially multiplies the initial value by the retained value percentage raised to the power of the number of years. This accurately models the compounding effect of depreciation over time.
To illustrate, let's consider the car's value after one year. Using the formula, we have:
Value after 1 year = $29,873 * (0.80)^1 = $23,898.40
After two years, the calculation becomes:
Value after 2 years = $29,873 * (0.80)^2 = $19,118.72
As you can see, the value decreases each year, and the formula captures this decreasing trend accurately. By understanding the components of the exponential decay formula, we can effectively model and predict the value of depreciating assets like Sunny's car. This knowledge is crucial for financial planning, insurance assessments, and understanding the long-term cost of ownership.
Now, let's examine the expressions provided and see how they relate to the concept of exponential decay and Sunny's car depreciation. We have two expressions to consider:
Our goal is to identify the expression that correctly models the car's value after t years, considering the 20% annual depreciation rate. To do this, we need to carefully analyze each expression and compare it to the exponential decay formula we discussed earlier.
The first expression, , might seem intuitive at first glance. It includes the initial value () and the depreciation rate (0.20). However, the crucial mistake here is that it uses the depreciation rate (0.20) directly as the base of the exponent. This implies that the car's value is decreasing exponentially based on the depreciation rate itself, which is incorrect. In exponential decay, we need to use the retained value, which is (1 - Depreciation Rate), as the base.
The second expression, , is even further from the correct representation. It uses 20 as the base of the exponent, which is not a percentage or a rate but a whole number. This would result in an exponentially increasing value, which contradicts the concept of depreciation. A value multiplied by 20 raised to the power of the number of years would rapidly increase, making this expression completely unsuitable for modeling depreciation.
Both expressions fail to accurately represent the exponential decay of the car's value. They either misuse the depreciation rate or completely misunderstand the concept of retained value. To find the correct expression, we need to remember the core principle of exponential decay: we need to multiply the initial value by the retained value (1 - Depreciation Rate) raised to the power of t. This ensures that the value decreases appropriately over time, reflecting the depreciation.
After analyzing the given expressions, we can confidently identify the correct one by applying the exponential decay formula. Remember, the formula is:
Value after t years = Initial Value * (1 - Depreciation Rate)^t
In Sunny's case:
- Initial Value = $29,873
- Depreciation Rate = 20% or 0.20
- (1 - Depreciation Rate) = 1 - 0.20 = 0.80
Therefore, the correct expression should be:
This expression accurately represents the car's value after t years. The initial value is multiplied by 0.80 (the retained value) raised to the power of t, capturing the exponential decay due to depreciation.
Now, let's consider the implications of this expression. It allows us to calculate the car's value at any point in the future. For example, after 5 years (t = 5), the value would be:
Value after 5 years = $29,873 * (0.80)^5 = $9,770.12 (approximately)
This demonstrates the significant impact of depreciation over time. After just five years, Sunny's car has lost a considerable portion of its initial value.
The correct expression also highlights the importance of understanding exponential decay in real-world scenarios. Depreciation is a crucial factor in financial planning, insurance assessments, and investment decisions. By using the correct formula, we can make informed decisions about the long-term value of assets and plan accordingly. Understanding the implications of depreciation can help individuals and businesses manage their finances effectively and make sound investment choices. This expression is not just a mathematical formula; it's a tool for understanding the real-world financial impact of depreciation.
In conclusion, understanding depreciation and exponential decay is crucial for managing assets and making informed financial decisions. In the scenario of Sunny's new car, we explored how the car's value decreases by 20% annually and determined the correct expression to model this depreciation over time. We learned that the correct expression is , which accurately represents the car's value after t years.
We also discussed why the other expressions were incorrect, emphasizing the importance of using the retained value (1 - Depreciation Rate) as the base of the exponential function. This understanding is key to applying the concept of exponential decay to real-world problems.
By correctly modeling depreciation, we can predict the future value of assets, plan for replacement costs, and make sound investment decisions. The example of Sunny's car serves as a practical illustration of how mathematics can be used to understand and manage financial aspects of our lives. This exploration reinforces the significance of mathematical literacy in navigating everyday financial scenarios and making informed choices.