Modeling the Value of an Antique Car Over Time Using Exponential Functions

The value of many antique items, including cars, can be modeled using exponential functions. This article explores how to adapt the given function to a more accurate representation in terms of years, offering insights into the exponential growth of the car's value as it ages.

Introduction to Exponential Growth

Exponential functions are used to model phenomena that grow or decay at a rate proportional to their current value. For antique items, such as cars, the value often increases over time as the rarity and condition of the car improve with each passing year.

The Original Function and Its Interpretation

The original function given is V 100 1.16^d, where V represents the value in thousands of dollars, and d represents decades after the car was sold at an auction for $110,000.

Converting Decades to Years

First, let's convert the function from decades to years. Since a decade consists of 10 years, we can express d in terms of t, where t represents the number of years:

d frac{t}{10}

Substituting this into the original equation, we get:

V 100 1.16^{frac{t}{10}}

To make this more explicit, we can express the function in terms of a base of e:

V 100 e^{ln(1.16) cdot frac{t}{10}}

Calculating ln(1.16):

ln(1.16) approx 0.148

Thus, the function can be simplified to:

V 100 e^{0.148 cdot frac{t}{10}} 100 e^{0.0148t}

Final Exponential Function in Terms of Years

The final exponential function that models the value of the antique car after t years is:

V_t 100 1.16^{frac{t}{10}} text{or equivalently} V_t 100 e^{0.0148t}

Both forms are valid and can be used depending on your preference.

Assumptions and Adjustments

Suppose the value of the car increases at a constant rate of 16% per decade. To make the function more accurate, we can adjust it to reflect a gradual increase:

V 110 cdot 1.015^t

This adjustment is based on the assumption that the car's value increases by 1.5% annually, which would equate to a 16% growth over a decade.

Example and Validation

Let's validate the model by comparing it to the given values:

① At t 0 (moments before being auctioned) the antique car was worth:

V 100 1.16^0 100k

It was first auctioned for $110,000, ten years later:

V 100 1.16^{frac{10}{10}} 100 cdot 1.16 116k

To check the accuracy, let's see how the model performs:

1. Appreciation within 32 years:

V 100 1.16^{frac{t}{10}}

110k 100 1.16^{frac{10}{10}} 116k

frac{110}{100} cdot 100^m 110k

ln(1.10) cdot m 2m 2.3010

m 1.1505

frac{1.1505}{0.003223} approx 35.74 text{years}

2. Doubling the value in approximately 42 years:

200 100 1.16^{20}

ln(2) cdot 20 cdot 0.0148t 42.8 text{years}

These calculations show that the model accurately captures the appreciation of the antique car's value over time.

In conclusion: The exponential function V_t 100 1.16^{frac{t}{10}} text{or equivalently} V_t 100 e^{0.0148t} effectively models the value of an antique car over time, assuming a steady increase of 16% per decade. For a more accurate representation, adjustments can be made by incorporating the actual growth rate observed over the past decades.