Question

Lila puts $400.00 into an account to use for school expenses. The account earns 3% interest, compounded continuously. How much will be in the account after 10 years?
Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.

Answer 1

To calculate the amount in Lila's account after 10 years using the formula for continuous compounding, we can use the equation:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the final amount,
- \( P \) is the principal (initial amount) = $400.00,
- \( r \) is the interest rate (as a decimal) = 3% = 0.03,
- \( t \) is the time in years = 10,
- \( e \) is the base of natural logarithms (approximately 2.71828).

Now we can substitute these values into the formula:

\[
A = 400e^{0.03 \cdot 10}
\]

Calculating \( 0.03 \cdot 10 \):

\[
0.03 \cdot 10 = 0.3
\]

Now, we substitute this back into the equation:

\[
A = 400e^{0.3}
\]

Next, we need to calculate \( e^{0.3} \):

\[
e^{0.3} \approx 1.34986
\]

Now, we can multiply this by 400:

\[
A \approx 400 \cdot 1.34986 \approx 539.944
\]

Finally, rounding to the nearest cent, we have:

\[
A \approx 539.94
\]

Thus, the amount in the account after 10 years will be **$539.94**.