Final answer:
It will take approximately 12.59 years for an initial investment of $20,000 to grow to $30,000, assuming a continuous compound interest rate of 16%.
Step-by-step explanation:
To calculate how many years it will take for an initial investment of $20,000 to grow to $30,000 with a continuous compound interest rate of 16%, we use the formula for continuous compounding, which is A = Pert, where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (decimal).
t is the time in years.
e is the base of the natural logarithm, approximately equal to 2.71828.
We know A = $30,000, P = $20,000, r = 0.16, and e is a constant. We need to solve for t:
30000 = 20000 * e(0.16t)
Divide both sides by 20000:
1.5 = e(0.16t)
Take the natural logarithm of both sides:
ln(1.5) = ln(e(0.16t))
Using the property of logarithms, we simplify:
ln(1.5) = 0.16t * ln(e)
Since ln(e) = 1:
ln(1.5) = 0.16t
Now, solve for t:
t = ln(1.5) / 0.16 ≈ 2.014 / 0.16 ≈ 12.59 years
Therefore, it will take approximately 12.59 years for the investment to grow from $20,000 to $30,000 at a 16% continuous compound interest rate.