An initial investment of $9000 grows at an annual interest rate of 5% compounded continuously. How long will it take to double the investment?
To find the time it takes for an investment to double with continuous compounding, we can use the formula:
t = ln(2) / (r * ln(1 + r))
Where:
t = time (in years)
r = annual interest rate (as a decimal)
In this case, the initial investment is $9000 and the annual interest rate is 5% (or 0.05 as a decimal). Let's plug these values into the formula:
t = ln(2) / (0.05 * ln(1 + 0.05))
Calculating this expression, we get:
t ≈ 13.86 years
Therefore, it will take approximately 13.86 years to double the initial investment of $9000 at an annual interest rate of 5% compounded continuously.
To calculate the time it takes to double an investment with continuous compounding, we can use the formula:
t = ln (A/P) / r
Where:
t = time in years
A = final amount or desired investment value
P = initial investment
r = annual interest rate
In this case, the initial investment (P) is $9000, and we want to find how long it takes to double the investment, which means the final amount (A) will be $18,000. The annual interest rate (r) is 5%.
Now let's substitute these values into the formula:
t = ln (18,000 / 9,000) / 0.05
To solve this, we can use a calculator or a math software that has a natural logarithm (ln) function. The natural logarithm of a number can be calculated by entering "ln" followed by the number.
t ≈ ln (2) / 0.05
Using a calculator or math software:
t ≈ 13.8629
Therefore, it will take approximately 13.8629 years to double the initial investment of $9000 at a 5% annual interest rate with continuous compounding.