Questions: Investments increase exponentially by about 15% every 2 years. If you start with a 500 investment, how much money would you have after 40 years?

Investments increase exponentially by about 15% every 2 years. If you start with a 500 investment, how much money would you have after 40 years?

Transcript text: Investments increase exponentially by about 15% every 2 years. If you start with a $500 investment, how much money would you have after 40 years?

Solution

Solution Steps

To solve this problem, we need to calculate the future value of an investment that grows exponentially. The formula for exponential growth is given by \( A = P(1 + r)^t \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial investment), \( r \) is the annual growth rate, and \( t \) is the time in years. In this case, the growth rate is 15% every 2 years, so we need to adjust the rate and time accordingly.

Step 1: Define the Variables

Let \( P = 500 \) be the initial investment. The growth rate every 2 years is given as \( 15\% \), which can be expressed as \( 0.15 \).

Step 2: Calculate the Annual Growth Rate

To find the annual growth rate \( r \), we convert the biannual growth rate to an annual rate: \[ r = (1 + 0.15)^{\frac{1}{2}} - 1 \approx 0.0724 \]

Step 3: Determine the Time Period

The total time period \( t \) is given as \( 40 \) years.

Step 4: Calculate the Future Amount

Using the formula for exponential growth: \[ A = P(1 + r)^t \] Substituting the values: \[ A = 500(1 + 0.0724)^{40} \] Calculating this gives: \[ A \approx 8183.2687 \]

Final Answer

The amount of money after 40 years is approximately \\(\boxed{8183.27}\\).

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