To determine the amount of money in the account after one year using the continuously compounding formula \( P(t) = P_0 e^{rt} \), where:
- \( P_0 \) is the initial principal (in this case, $1,200),
- \( r \) is the annual interest rate (0.16 for 16%),
- \( t \) is the time in years (1 year),
- \( e \) is the base of the natural logarithm (approximately 2.71828).
Plugging in the values:
\[ P(1) = 1200 e^{0.16 \times 1} \]
Calculating the exponent:
\[ 0.16 \times 1 = 0.16 \]
Now substituting back into the formula:
\[ P(1) = 1200 e^{0.16} \]
Next, calculate \( e^{0.16} \):
\[ e^{0.16} \approx 1.17351 \quad (\text{using a calculator or mathematical software}) \]
Now, substituting this back into the equation:
\[ P(1) \approx 1200 \times 1.17351 \] \[ P(1) \approx 1408.212 \]
Therefore, the amount of money in the account after one year:
\[ P(1) \approx 1408.21 \]
So the final amount in the account after one year is $1,408.21.