Question

Lacey is saving up money to buy a car. Lacey puts $6,000.00 into an account which earns 3% interest, compounded continuously. How much will she have in the account after 2 years?
Round your answer to the nearest cent.

Answer 1

To calculate the amount in Lacey's account after 2 years with continuous compounding interest, we use the formula for continuous compounding:

\[
A = Pe^{rt}
\]

where:
- \( A \) = the amount of money accumulated after \( t \) years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (decimal).
- \( t \) = the number of years the money is invested or borrowed for.
- \( e \) = the base of the natural logarithm, approximately equal to 2.71828.

Given:
- \( P = 6000 \)
- \( r = 0.03 \)
- \( t = 2 \)

Now, substituting in the values:

\[
A = 6000 \cdot e^{0.03 \cdot 2}
\]

Calculating \( 0.03 \cdot 2 \):

\[
0.03 \cdot 2 = 0.06
\]

Now we calculate \( e^{0.06} \):

\[
e^{0.06} \approx 1.061837
\]

Next, we can calculate \( A \):

\[
A = 6000 \cdot 1.061837 \approx 6367.022
\]

Finally, rounding to the nearest cent:

\[
A \approx 6367.02
\]

Thus, after 2 years, Lacey will have approximately **$6,367.02** in her account.