Questions: Adriana Lima invests 8,175 into an account that earns 7.9% compounded continuously. If she checks her account 14 years later, what will the balance in her account be?

Solution Steps

To solve this problem, we need to use the formula for continuous compounding, which is given by \( A = P \times e^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest. \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (in decimal), \( t \) is the time the money is invested for in years, and \( e \) is the base of the natural logarithm.

1. Identify the principal amount \( P = 8175 \).
2. Convert the annual interest rate from a percentage to a decimal \( r = 7.9/100 = 0.079 \).
3. Identify the time period \( t = 14 \) years.
4. Use the continuous compounding formula to calculate the future value.

We are given the following values:

• Principal amount \( P = 8175 \)
• Annual interest rate \( r = 0.079 \)
• Time period \( t = 14 \)

The formula for continuous compounding is given by:

\[ A = P \times e^{rt} \]

Substituting the known values into the formula:

\[ A = 8175 \times e^{0.079 \times 14} \]

Calculating the exponent:

\[ 0.079 \times 14 = 1.106 \]

Now, substituting this back into the equation:

\[ A = 8175 \times e^{1.106} \]

Calculating \( e^{1.106} \):

\[ e^{1.106} \approx 3.017 \]

Now, substituting this value into the equation for \( A \):

\[ A \approx 8175 \times 3.017 \approx 24706.8545 \]

Final Answer