The problem in the image asks how much money needs to be deposited now to obtain $600 in 5 years if the interest rate is 3% compounded continuously.

This problem uses the formula for continuous compounding:

$A = Pe^{rt}$

Where:

• $A$ is the final amount ($600),
• $P$ is the principal amount to be deposited (unknown),
• $r$ is the interest rate (3% or 0.03),
• $t$ is the time in years (5 years),
• $e$ is the base of the natural logarithm (approximately 2.718).

We rearrange the formula to solve for $P$:

$P = \frac{A}{e^{rt}}$

Substitute the given values: $P = \frac{600}{e^{0.03 \cdot 5}}$

Calculate $P$:

1. $r \cdot t = 0.03 \cdot 5 = 0.15$,
2. $e^{0.15} \approx 1.161834$,
3. $P = \frac{600}{1.161834} \approx 516.97$.

Thus, the initial deposit required is approximately $516.97.

Would you like me to explain this calculation in more detail or solve related problems?

Here are five related questions:

1. How does the formula change if the interest is compounded monthly instead of continuously?
2. What happens to the required initial deposit if the interest rate increases?
3. What would the final amount be after 10 years instead of 5?
4. How can you derive the continuous compounding formula from simpler interest equations?
5. What is the effect of doubling the principal on the final amount in continuous compounding?

Tip: Always double-check your rounding when working with financial equations to ensure precision!