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How many years will it take for an initial investment of $20,000 to grow to $30,000 ? Assume a rate of interest of 9% compounded continuously.

It will take about ____ years for the investment to grow to $30,000. (Round to two decimal places as needed.)

User Dolo
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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.

User Sukesh Chand
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