Complete the table assuming continuously compounded interest. (Round your answers to two decimal places.)
Initial Investment== ?
Annual % Rate== ?
Time to Double== 11YRS
Amount After 10 Years== $1600
Using formula A=Pe^rt ?
2 = e^(11r)
r = ln(2) / 11
1600 = P e^[10 ln(2) / 11]
Thank you so much!
To complete the table, we need to find the missing values: the initial investment and the annual interest rate. We are given the time to double and the amount after 10 years.
The formula for continuously compounded interest is given by A = Pe^rt, where:
A = the final amount
P = the principal (initial investment)
e = Euler's number, approximately 2.71828
r = the annual interest rate (in decimal form)
t = the time period (in years)
To find the missing values, let's start with the initial investment (P). We can rearrange the formula to solve for P:
P = A / e^(rt)
Given the amount after 10 years (A = $1600), we can substitute this value into the equation:
P = $1600 / e^(rt)
Next, we need to find the annual interest rate (r) to complete the table. We can use the formula to find the doubling time and solve for r. The formula to calculate the time to double is:
t = ln(2) / r
Given the time to double is 11 years, we can rearrange this formula to solve for r:
r = ln(2) / t = ln(2) / 11
Using a calculator, we find that ln(2) is approximately 0.69315.
Now, substituting the values we have into the formula for P:
P = $1600 / e^((ln(2)/11) * 10)
Simplifying this expression further:
P ≈ $1600 / e^(10 * 0.69315 / 11)
Calculating e^(10 * 0.69315 / 11) ≈ 1.75349, we find:
P ≈ $1600 / 1.75349 ≈ $912.80
Therefore, the initial investment is approximately $912.80.
Now, to find the annual interest rate (r):
r ≈ ln(2) / 11 ≈ 0.0631
Converting this value to a percentage, we have an annual interest rate of approximately 6.31%.
Completing the table:
Initial Investment: $912.80
Annual % Rate: 6.31%
Time to Double: 11 years
Amount After 10 Years: $1600