The formula for continuously compounded interest is:

$A(t) = ae^{rt}$

Where:

• $A(t)$ is the value of the investment after time $t$,
• $a$ is the initial value of the investment,
• $r$ is the interest rate (expressed as a decimal),
• $t$ is the time in years,
• $e$ is the base of the natural logarithm (approximately $e \approx 2.718281$).

We are given:

• $a = 1000$ (initial value),
• $r = 0.08$ (8% interest rate),
• $t = 50$ years.

Now, we can plug these values into the formula:

$A(50) = 1000 \times e^{(0.08 \times 50)}$ $A(50) = 1000 \times e^{4}$

Next, using the precise value of $e \approx 2.718281$, we calculate:

$e^4 \approx 2.718281^4 \approx 54.59815$

Now, multiply by the initial amount:

$A(50) = 1000 \times 54.59815 \approx 54,598.15$

Thus, the value of the investment after 50 years is approximately $54,598.15.

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Would you like further details on this calculation or have other questions? Here are 5 related questions to explore:

1. How does the interest rate impact the value of the investment over time?
2. What would happen if the time period was shortened to 25 years?
3. How does continuously compounded interest compare to annually compounded interest?
4. If the investment rate was increased to 10%, what would the final amount be?
5. How can the formula be applied to real-life financial decisions?

Tip: To understand how exponential growth works in finance, it's helpful to visualize the growth curve. As the compounding period increases, the effect of the interest rate becomes significantly more pronounced.