Problem 33 - College Algebra and Calculus: An Applied Approach

Question

Suppose that the value of a piece of property doubles every 15 years. If you buy the property for \(\$ 64,000\), its value \(t\) years after the date of purchase should be \(V(t)=64,000(2)^{2 / 15} .\) Use the model to approximate the value of the property (a) 5 years and (b) 20 years after it is purchased.

Step-by-Step Solution

Answer

The approximate value of the property 5 years after purchase is $76,598.26 and 20 years after purchase is $107,374.65.

Step 1: Understand the exponential model

The given model is \(V(t)=64,000(2)^{t / 15},\) where \(t\) is the time period in years. This model signifies that the property value, \(V(t)\), doubles every 15 years.

Step 2: Calculate the approximate value 5 years after purchase

Substitute \(t=5\) into \(V(t)=64,000(2)^{t / 15} .\) We get \(V(5)=64,000(2)^{5 / 15} .\) Calculate this value to find the property value 5 years after purchase.

Step 3: Calculate the approximate value 20 years after purchase

Substitute \(t=20\) into \(V(t)=64,000(2)^{t / 15} .\) We get \(V(20)=64,000(2)^{20 / 15} .\) Calculate this value to find the property value 20 years after purchase.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model many real-world phenomena, particularly growth or decay processes. They're characterized by the formula
\[ f(x) = ab^{x} \] where:

• \( a \) is the initial value,
• \( b \) is the base or growth factor,
• \( x \) represents time or another independent variable.

In our example, the property value is modeled by \[ V(t)=64,000(2)^{t / 15} \] Here, \( 64,000 \) is the initial value (or purchase price), \( 2 \) is the growth factor indicating the property value doubles, and \( t \) stands for the time in years. To determine the property's value at any given time, you plug the number of years, \( t \), into the model and solve for \( V(t) \).Exponential growth models like this are incredibly useful for predicting future scenarios where change occurs at a rate proportional to the current value—such as population growth, radioactive decay, and, as demonstrated here, property value appreciation over time.

Property Value Appreciation

Property value appreciation refers to the increase in the value of real estate over time. It's influenced by various factors, including economic conditions, location, property improvements, and inflation. The model we're considering uses an exponential function to express this appreciation, which simplifies these factors into a consistent rate of growth.

Understanding the Model

In the exponential growth model provided in the exercise (
\[ V(t)=64,000(2)^{t / 15} \]), the doubling rate is encoded in the formula. This mirrors the nature of property investment where, ideally, the investor seeks to purchase assets that will appreciate in value at an exponential rate. It's a simplified model and doesn't take into account real-world complexities like market fluctuations, but it serves well for understanding the fundamental concepts of exponential appreciation. When assuming a regular growth pattern, like a property doubling every 15 years, the expression captures how much more valuable your property could be after any given period.

Calculating Future Value

Calculating future value is essential for forecasting the worth of investments over time. In our exercise, future value is determined using an exponential function, predicting the property's value in 5 years and 20 years.

Step-by-Step Calculation

To calculate the property value after 5 years, you would substitute 5 for \( t \) in the provided formula, resulting in:
\[ V(5)=64,000(2)^{5 / 15} \]. Similarly, to find the approximate value after 20 years, substitute 20 for \( t \):
\[ V(20)=64,000(2)^{20 / 15} \]. Each calculation provides an estimate of future value based on the assumed continuous exponential rate of growth. Understanding how to perform these calculations is crucial for anyone engaged in financial planning or investment, especially in real estate. It allows for informed decision-making and helps individuals evaluate the potential outcomes of their investments over time.