To calculate the amount Sam will have in his account after four years, we use the formula for continuously compounded interest:

$A = P e^{rt}$

Where:

• $A$ is the amount in the account after time $t$.
• $P$ is the principal (initial deposit).
• $r$ is the annual interest rate (expressed as a decimal).
• $t$ is the time in years.
• $e$ is Euler's number (approximately 2.71828).

Given:

• $P = 1500$ (initial deposit).
• $r = 5\% = 0.05$ (annual interest rate).
• $t = 4$ years.

Now, plug in the values:

$A = 1500 \times e^{0.05 \times 4}$

$A = 1500 \times e^{0.20}$

We can now calculate the value of $e^{0.20}$.

$A \approx 1500 \times 1.2214$

$A \approx 1832.10$

Result:

Sam will have approximately $1832.10 in the account after four years.

Would you like a more detailed explanation or have any further questions?

Related Questions:

1. How would the amount change if the interest rate was 6% instead of 5%?
2. What is the difference between continuous compounding and annual compounding?
3. How long would it take for Sam’s deposit to double with continuous compounding at 5% interest?
4. How much more would Sam earn after 4 years if the interest was compounded monthly instead of continuously?
5. How would the account balance change if Sam made an additional deposit of $500 after two years?

Tip: When using continuous compounding, the formula $A = P e^{rt}$ provides the most accurate result. However, for less frequent compounding, you can use the formula for compound interest with the number of periods per year.