Question

You buy a used truck in 1996 for \(\$ 10,500\). Each year the truck depreciates by \(10 \% .\) Write an exponential decay model to represent this situation. Then estimate the value of the truck in 10 years.

Short answer

The estimated value of the truck after 10 years, using the derived exponential decay model is \$ V = 10500 \times 0.9^{10} = \approx $ 3643.98.

Step-by-step solution

Understand the Exponential Decay Model

Exponential decay is given by the formula: \( V = P(1 - r)^t \), where: V is the final value, P is the initial principal amount, r is the decay rate, and t is time.

Formulate the Depreciation Model

We are given that the initial value P is $10,500, the decay rate r is 10% or 0.1, and the time t is in years. Substitute these values into the formula to get the model: \( V = 10500 \times (1 - 0.1)^t \), which simplifies to \( V = 10500 \times 0.9^t \). This is our depreciation model.

Estimate the Value after 10 Years

Using the model, we can estimate the value of the truck after 10 years. Substituting \( t = 10 \) into the exponential decay model gives us \( V = 10500 \times 0.9 ^ {10} \). Calculate this to determine the value of V.

Key concepts

Exponential Decay

The concept of exponential decay refers to the process by which an initial amount decreases over time at a rate proportional to its current value. This pattern of decline is prevalent in nature and economics, seen in radioactive decay, population decline, and asset depreciation.
An exponential decay model is mathematically represented by the formula: \[ V = P(1 - r)^t \]
Where:

• \( V \) is the final value after time \( t \),
• \( P \) is the initial principal amount (also considered the starting value),
• \( r \) represents the rate of decay (expressed as a decimal), and
• \( t \) denotes time, often measured in consistent intervals like years or days.

Understanding this model is crucial for predicting how quantities diminish over a period when subjected to consistent percentage decreases.

Depreciation Model

A depreciation model is a specific application of exponential decay, often used to estimate the decrease in value of an asset, such as machinery, vehicles, or equipment over time. In accounting and finance, depreciation models are crucial for understanding assets' value loss, budgeting, and tax calculations.
The model relies on the fundamental principle of exponential decay to predict the future value of an asset given its current worth, depreciation rate, and the time period considered. The formula to gauge the depreciated value is a variation of the exponential decay formula where the rate of decay represents the yearly depreciation percentage:
\[ V = P(1 - r)^t \]
In this context, planning for asset replacement and financial strategizing becomes more efficient since businesses can anticipate when an asset will lose its utility or fall below a certain value threshold.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations accurately. For exponential decay, the model helps predict the value of a decreasing quantity over time. In the case of the truck in our problem, the depreciation model is established by identifying the initial purchase price, the constant rate of depreciation, and the duration of usage.
In the provided solution, the equation \( V = 10500 \times (1 - 0.1)^t \) symbolizes the truck’s worth through the years. This formula is the mathematical model that encapsulates the vehicle's yearly 10% depreciation. By inputting different values of \( t \), we can forecast the truck’s value at any future point in time. It's an excellent example of using mathematical models to make informed decisions, such as budgeting for replacements or gauging when to sell the asset.

Percentage Decrease

The percentage decrease is a means to measure the reduction in value or quantity, expressed as a percentage of the original amount. This concept is an intrinsic part of exponential decay models, indicating how swiftly the value depreciates each period. It is calculated by dividing the decrease in value by the original value and then multiplying by 100 to get a percentage.
In our exercise, the truck's value dips by 10% annually. This percentage is critical to formulating the depreciation model since it determines the decay rate (\( r \) in the formula), impacting how rapidly the truck loses its value. Understanding percentage decrease aids students and professionals in sectors like finance and science to anticipate changes over time, enabling better planning and decision-making.