Gabriela invests $1,200 into a continuously compounding account with an annual interest rate of 16 percent.

Use the formula P(t)=1,200e
0.16t
to determine the amount of money in the account after one year. Include cents, if needed.
\ □

1 answer

To determine the amount of money in the account after one year using the continuously compounding interest formula, you can use the formula:

\[ P(t) = P_0 e^{rt} \]

Where:

  • \( P(t) \) is the amount of money in the account after time \( t \),
  • \( P_0 \) is the principal amount (initial investment),
  • \( r \) is the annual interest rate (in decimal),
  • \( t \) is the time in years,
  • \( e \) is the base of the natural logarithm (approximately 2.71828).

In this case:

  • \( P_0 = 1200 \)
  • \( r = 0.16 \) (16 percent as a decimal)
  • \( t = 1 \)

Now plug in the values:

\[ P(1) = 1200 e^{0.16 \times 1} \] \[ P(1) = 1200 e^{0.16} \] Now calculate \( e^{0.16} \):

Using a calculator, we find:

\[ e^{0.16} \approx 1.1735 \]

Now substitute this value back into the formula:

\[ P(1) = 1200 \times 1.1735 \approx 1408.20 \]

Thus, the amount of money in the account after one year is approximately $1,408.20.