To find the interest earned on a deposit with continuous compounding, we use the formula:

$A = P e^{rt}$

where:

• $A$ is the final amount,
• $P$ is the principal (initial amount) = $30,000,
• $r$ is the annual interest rate (as a decimal) = 1% = 0.01,
• $t$ is the time in years = 6 months = 0.5 years,
• $e$ is the base of the natural logarithm (approximately 2.71828).

First, calculate the final amount:

$A = 30,000 \times e^{0.01 \times 0.5}$

$A = 30,000 \times e^{0.005}$

Now let's compute this value.The final amount after 6 months is approximately $30,150.38. Therefore, the interest earned on the $30,000 deposit over six months at a 1% interest rate, compounded continuously, is approximately $150.38.

Would you like more details or have any questions about this calculation?

Here are 5 related questions to expand your understanding:

1. How does continuous compounding differ from simple and regular compounding?
2. What would the interest be if it were compounded annually instead?
3. How would the interest change if the interest rate were 2% instead of 1%?
4. What happens to the interest earned if the deposit time is extended to one year?
5. How can you use the natural logarithm $e$ in other financial calculations?

Tip: Continuous compounding can result in slightly higher interest than standard compounding methods, especially over longer periods or at higher rates.