Problem 49 - Precalculus

Question

Suppose an investor buys land in a rural area for \(\$ 1,500\) an acre and sells some of it 5 years later at \(\$ 3,000\) an acre and the rest of it 10 years later at \(\$ 6,000 .\) Write a function that models the value of land in that area, assuming the growth rate stays the same. What would the expected cost per acre be 30 years after the initial investment of \(\$ 1,500 ?\)

Step-by-Step Solution

Answer

The expected cost per acre after 30 years is approximately \(\$34,655.55\).

Step 1: Define the Objective

We need to write a function that models the growth of the value of land over time and estimate its value 30 years after an initial investment.

Step 2: Understand the Given Information

The land is initially bought for \(\\(1,500\) per acre. Some is sold at \(\\)3,000\) per acre after 5 years and the rest at \(\$6,000\) per acre after 10 years.

Step 3: Determine the Growth Rate

The value of land doubles from \(\\(1,500\) to \(\\)3,000\) in 5 years. Using this information, we can calculate the annual growth rate \( r \) using the formula for compound interest: \[ V(t) = P \times (1 + r)^t \] where \( V(t) \) is the value at time \( t \), \( P \) is the principal, and \( r \) is the growth rate. After 5 years, the equation is: \[ 3000 = 1500 \times (1 + r)^5 \]

Step 4: Calculate the Growth Rate

Solving the equation from the previous step: \[ 3000 = 1500 \times (1 + r)^5 \] Divide both sides by 1500: \[ 2 = (1 + r)^5 \] Taking the 5th root on both sides: \[ 1 + r = 2^{1/5} \] \[ r = 2^{1/5} - 1 \] Using a calculator, \( r \approx 0.1487 \) or approximately 14.87% annual growth.

Step 5: Write the Function

With the growth rate calculated, the value of the land per acre as a function of time \( t \) in years is: \[ V(t) = 1500 \times (1 + 0.1487)^t \]

Step 6: Evaluate the Function for 30 Years

To find the expected value per acre after 30 years, we substitute t = 30 into the function: \[ V(30) = 1500 \times (1.1487)^{30} \] Calculate this using a calculator: \[ V(30) \approx 1500 \times 23.1037 = 34655.55 \]

Key Concepts

Compound Interest

Compound interest is a powerful financial concept that involves the growth of an investment over time, not just based on the initial amount invested (known as the principal), but also on the accumulated interest from previous periods. In simple terms, it means that your interest begins to earn interest.
One of the key formulas for compound interest is given by:

• \[ V(t) = P \times (1 + r)^t \]

Here, \( V(t) \) represents the future value of the investment after \( t \) years, \( P \) is the principal amount initially invested, \( r \) is the annual growth rate, and \( t \) represents the time period in years.
Compound interest is what allows the initial amount to grow at an increasing rate, and is crucial in understanding long-term investments.

Growth Rate Calculation

Determining the growth rate is essential for modeling how an investment grows over time. In the given exercise, we calculate the growth rate using the formula associated with compound interest.
The process involves solving the equation that uses both the initial value and the value after a certain period to find \( r \), the annual growth rate.
Based on the problem, after 5 years, the land's value doubles from \( \\(1,500 \) to \( \\)3,000 \). Using the compound interest formula, the equation becomes:

• \[ 3000 = 1500 \times (1 + r)^5 \]

This can be simplified by dividing both sides by 1500, resulting in:

• \[ 2 = (1 + r)^5 \]

Solving for \( r \), you take the 5th root of both sides and subtract 1:

• \[ r = 2^{1/5} - 1 \]

This gives us an annual growth rate \( r \approx 0.1487 \) or 14.87%. This rate describes how quickly and exponentially the investment grows each year.

Function Modeling

Function modeling is the process of creating equations to represent real-world phenomena. In this exercise, it is used to model the changing value of land over time.
After determining the growth rate, we can construct a function that describes how the investment evolves.
The function utilized in this scenario is:

• \[ V(t) = 1500 \times (1.1487)^t \]

This function helps us predict the land's price at any point in time. By substituting 30 for \( t \), you can find the expected value of the land after 30 years:

• \[ V(30) = 1500 \times (1.1487)^{30} \approx 34655.55 \]

Therefore, function modeling offers a powerful tool for forecasting and understanding potential future outcomes of investments.