Questions: Suppose that 16,784 is invested at an interest rate of 6.7% per year, compounded continuously. a) Find the exponential function that describes the amount in the account after time t, in years. b) What is the balance after 1 year? 2 years? 5 years? 10 years? c) What is the doubling time?

Solution Steps

The exponential function that describes the amount in the account after time \( t \), in years, is given by:

\[ P(t) = 16784 e^{0.067 t} \]

Using the exponential function, we can calculate the balance after 1, 2, 5, and 10 years:

• After 1 year: \[ P(1) \approx 17947.06 \]
• After 2 years: \[ P(2) \approx 19190.71 \]
• After 5 years: \[ P(5) \approx 23463.03 \]
• After 10 years: \[ P(10) \approx 32799.92 \]

The doubling time \( T \) can be calculated using the formula:

\[ T = \frac{\ln(2)}{r} \approx 10.35 \text{ years} \]

Final Answer