Final answer:
Using the formula for continuous compound interest and solving for time, it will take approximately 5.56 years for a $20,000 investment to grow to $30,000 at a rate of 9% interest compounded continuously.
Step-by-step explanation:
To determine how many years it will take for an initial investment of $20,000 to grow to $30,000 with a rate of interest of 9% compounded continuously, we use the formula for continuous compounding, 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.
We are given that A=$30,000, P=$20,000, and r=0.09. Plugging these values into the formula and solving for t, we have:
30,000 = 20,000e(0.09)t
To find t, we need to isolate it:
\(\frac{30,000}{20,000} = e(0.09)t\)
1.5 = e(0.09)t
Now take the natural logarithm of both sides:
ln(1.5) = ln(e(0.09)t)
ln(1.5) = (0.09)t\(\cdot\ln(e)
Since \(\ln(e) = 1
ln(1.5) = 0.09t
\(\frac{ln(1.5)}{0.09} = t\)
t approximately equals 5.56 years after calculating the numerator and dividing by 0.09
Therefore, it will take about 5.56 years for the investment to grow to $30,000 when rounded to two decimal places.