The question you're working on involves calculating the amount in a savings account after 18 years, with interest compounded continuously.

The formula for continuous compounding is:

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

where:

• $A$ is the amount after time $t$,
• $P$ is the principal amount (initial deposit),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time the money is invested (in years),
• $e$ is the base of the natural logarithm (approximately 2.71828).

From the image:

• $P = 30000$,
• $r = 5\% = 0.05$,
• $t = 18$ years.

Now, we calculate $A$:

$A = 30000 \cdot e^{0.05 \cdot 18}$

$A = 30000 \cdot e^{0.9}$

$A = 30000 \cdot 2.4596 \quad (\text{using} \, e^{0.9} \approx 2.4596)$

$A \approx 30000 \cdot 2.4596 = 73788$

Rounding to the nearest cent:

$A \approx 73788.08$

So, the correct amount after 18 years should be $73,788.08, not $73,787.96 as shown.

Would you like any further clarifications or detailed steps?

Here are 5 related questions:

1. How does continuous compounding differ from annual compounding?
2. What would be the total amount if the interest rate were 6% instead of 5%?
3. How would you calculate the amount after 25 years instead of 18 years?
4. What is the formula for compound interest when it's compounded annually instead of continuously?
5. How would changes in the principal amount affect the final balance?

Tip: Continuous compounding grows your money faster because it assumes the interest is added instantly and continuously, unlike annual or monthly compounding.