Final answer:
It will take approximately 11.57 years for an initial investment of $10,000 to grow to $5,000 with a compound interest rate of 6% compounded continuously.
Step-by-step explanation:
To find out how many years it will take for an initial investment of $10,000 to grow to $5,000 with a compound interest rate of 6% compounded continuously, we can use the formula:
A = P * e^(rt)
Where A is the final amount, P is the initial investment, e is Euler's number (~2.71828), r is the interest rate, and t is the number of years.
Plugging in the given values:
5000 = 10000 * e^(0.06t)
Dividing both sides by 10000:
0.5 = e^(0.06t)
Taking the natural logarithm of both sides:
ln(0.5) = ln(e^(0.06t))
Using the logarithmic property: ln(e^x) = x
ln(0.5) = 0.06t
Dividing both sides by 0.06:
t = ln(0.5)/0.06
Using a calculator, we find that t ≈ 11.57 years.