Problem 72 - College Algebra with Modeling and Visualization

Question

Lake Property In some states, lake shore property is increasing in value by \(15 \%\) per year. Determine the value of a \(\$ 90,000\) lake lot after 5 years.

Step-by-Step Solution

Answer

The lake lot will be worth approximately \( \$181,022.13 \) after 5 years.

Step 1: Identify Initial Value and Growth Rate

The initial value of the lake lot is given as \( \$90,000 \). The annual growth rate for the property is \( 15\% \), which we'll express as a decimal: \( 0.15 \).

Step 2: Use the Compound Interest Formula

The value of an investment growing at a constant rate can be calculated using the compound interest formula: \[ A = P(1 + r)^n \]where:- \( A \) is the amount after time \( n \),- \( P \) is the principal amount \( \$90,000 \),- \( r \) is the annual interest rate as a decimal \( 0.15 \),- \( n \) is the number of years, which is 5 in this case.

Step 3: Substitute Values into the Formula

Plug the known values into the formula: \[A = 90000(1 + 0.15)^5\]

Step 4: Calculate Inside the Parenthesis

Calculate the expression inside the parenthesis: \(1 + 0.15 = 1.15\)

Step 5: Raise to the Power of 5

Raise 1.15 to the 5th power: \(1.15^5 \approx 2.011357\)

Step 6: Multiply by the Initial Value

Multiply the result by the initial value to find the future value of the property:\[A = 90000 \times 2.011357 \approx 181,022.13\]

Step 7: Final Answer

The value of the lake lot after 5 years is approximately \( \$181,022.13 \).

Key Concepts

Investment Growth

Investment growth is a crucial concept when discussing finances and future planning. It refers to the increase in value of an investment over time. In many cases, like with the lake property, this growth is fueled by factors such as market demand, location, and economic conditions.
One of the key drivers of investment growth is the annual growth rate, which explains how much the investment will grow each year.
For the lake property, the value increases by 15% per year. This increase is referred to as compounded, meaning it applies to both the original investment value and the accumulated interest from previous years.

• Understanding investment growth helps make smart financial decisions.
• It emphasizes the importance of time in growing wealth effectively.
• Recognizing growth rates assists in predicting future value.

Comprehending the growth rate and the factors influencing it can offer great insights into the potential that holds in various investments over time.

Future Value Calculation

The future value calculation is the process used to determine what a current investment is worth in the future, considering a specified interest rate and time period. This is crucial for making informed investment decisions.
The future value formula is: \[ A = P(1 + r)^n \]*where:*

• \( A \) is the future amount, or the value of the investment at the end of the specified time.
• \( P \) is the principal, or initial amount invested.
• \( r \) is the annual growth rate, expressed as a decimal.
• \( n \) is the number of compounding periods, often given in years.

For example, the lake lot priced at \( \\(90,000 \) at a 15% annual growth rate over 5 years becomes \( \\)181,022.13 \) when calculated using this formula.
Well understanding this process is invaluable for predicting the future value of investments.

Annual Growth Rate

The annual growth rate is a percentage that indicates how much an investment increases in value each year. It is pivotal in assessing how profitable an investment might be over time.
In the case of the lake shore property, the annual growth rate is 15%. This rate reflects how much the investment is expected to grow each year based on past market conditions.
An annual growth rate can be easily converted from a percentage to a decimal by simply dividing by 100.

• A rate of 15% becomes 0.15 as a decimal.
• This conversion is important for using mathematical formulas.
• Understanding this conversion helps in applying it to different financial formulas effectively.

Grasping the annual growth rate lets investors calculate potential returns and make strategic decisions based on expected future values.