Problem 71 - Precalculus with Limits

Question

Depreciation After \(t\) years, the value of a wheelchair conversion van that originally cost \(\$ 49,810\) depreciates so that each year it is worth \(\frac{7}{8}\) of its value for the previous year. (a) Find a model for \(V(t),\) the value of the van after \(t\) years. (b) Determine the value of the van 4 years after it was purchased.

Step-by-Step Solution

Answer

The value of the van 4 years after purchase is given by: \(V(4) = 49810 \times \left(\frac{7}{8}\right)^4\). By approximating, the value comes out to be approximately \$34640.02

Step 1: Establish the depreciation model

The value of the van decreases every year by a fraction \(\frac{7}{8}\) of its previous value. If the original value of the van is denoted as \(V_0\), the model that describes the value \(V(t)\) of the van after \(t\) years is given by: \[V(t) = V_0 \times \left(\frac{7}{8}\right)^t\] Here, \(V_0 = \$ 49,810\) is the initial cost of the van.

Step 2: Apply the model for \(t = 4\) years

To find the value of the van 4 years after it was purchased, substitute \(t = 4\) into the depreciation model obtained in Step 1. \[V(4) = 49810 \times \left(\frac{7}{8}\right)^4\]

Step 3: Evaluate the expression

Evaluate \[V(4) = 49810 \times \left(\frac{7}{8}\right)^4\] to find the value of the van 4 years after purchase.

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept that describes the process of a quantity decreasing rapidly at a consistent rate over time. This concept is commonly used in various fields, such as physics, biology, and finance. In the case of the wheelchair conversion van, the value depreciates by multiplying with a constant fraction every year. This fraction represents the remaining value as time passes.

• In our scenario, the fraction \( \frac{7}{8} \) indicates that the van's value retains 87.5% of its previous value each year.
• This decrease is exponential because, for each successive year, the reduction is a percentage of the current value, not the original value.

The formula for exponential decay is written as \[ V(t) = V_0 \times \left(\frac{7}{8}\right)^t \] where \( V_0 \) is the starting value, and \( t \) is the number of years.Exponential functions provide a powerful tool for modeling depreciation, allowing for precise predictions regarding the future value of assets.

Mathematical Modeling

Mathematical modeling is the practice of using mathematical language and concepts to represent real-world phenomena. This process involves forming equations that accurately describe how a particular situation behaves over time.
In our exercise, mathematical modeling is demonstrated through the depreciation of the wheelchair conversion van. We represent the changing value of the van with an exponential equation: \[ V(t) = V_0 \times \left(\frac{7}{8}\right)^t \]

• By defining \( V_0 \) as \$49,810,\ we create a model that can estimate the van's value at any given time \( t \).
• The power of this modeling lies in its ability to provide clear predictions based on observable patterns and constants (in this case, \( \frac{7}{8} \)).

The purpose of mathematical modeling is to simplify complex systems into comprehensible equations that help in analysis and decision-making. It serves as a bridge between theoretical mathematics and practical applications, making complex problems easier to tackle.

Financial Mathematics

Financial mathematics focuses on the application of mathematical methods to solve problems related to finance and economics. This area of mathematics is essential for making informed decisions regarding investments, depreciation, loans, and other financial matters.
The depreciation model used in the exercise is a classic example of financial mathematics. Here’s why:

• It provides an accurate method to estimate the loss in the value of an asset over time, crucial for accounting and financial planning.
• The model helps stakeholders determine the present and future worth of assets, enabling better resource management.

When we apply the model to determine the van's value after four years, as shown by \[ V(4) = 49810 \times \left(\frac{7}{8}\right)^4 \], we perform a calculation rooted in financial mathematics. This calculation serves practical needs such as asset valuation, tax preparation, and resale value estimation.
Overall, financial mathematics is a key tool in the realm of economics, aiding individuals and businesses alike to make rational decisions based on quantitative analysis.