Santiago Cox
04/10/2024 · Junior High School
You deposit \( \$ 4000 \) in an account earning \( 2 \% \) interest compounded continuously. How much will you have in the account in 10 years? \( \$ \)
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Tutor-Verified Answer
Step-by-step Solution
To calculate the amount in an account with continuous compounding interest, you can use the formula:
\[
A = Pe^{rt}
\]
where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (the initial deposit),
- \( r \) is the annual interest rate (in decimal),
- \( t \) is the time the money is invested for (in years),
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- \( P = 4000 \)
- \( r = 0.02 \) (which is \( 2\% \) expressed as a decimal)
- \( t = 10 \)
Now, plug in the values into the formula:
\[
A = 4000 \cdot e^{0.02 \cdot 10}
\]
Calculating the exponent:
\[
0.02 \cdot 10 = 0.2
\]
Now, calculate \( e^{0.2} \):
\[
e^{0.2} \approx 1.2214
\]
Now, substitute this back into the formula:
\[
A \approx 4000 \cdot 1.2214
\]
Calculating the final amount:
\[
A \approx 4885.60
\]
Thus, the amount in the account after 10 years will be approximately:
\[
\boxed{4885.60}
\]
Quick Answer
The amount in the account after 10 years will be approximately \( \$4885.60 \).
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