Final answer:
An initial investment of $2000 at a 4% annual interest rate compounded continuously will take approximately 17.33 years to double. We use the formula for continuous compounding, 2 = e^(rt), and solve for time (t) by taking the natural logarithm of both sides and dividing by the interest rate.
Step-by-step explanation:
To determine how long it will take for an initial investment to double when compounded continuously, 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, r is the annual interest rate, t is the time (years), and e is the base of the natural logarithm, approximately equal to 2.71828.
Since we want to double the investment, our equation becomes 2P = Pert.
The P cancels out, and we're left with 2 = ert.
We must solve for t. Taking the natural logarithm of both sides gives us ln(2) = rt.
The time it takes to double can be found by dividing the natural log of 2 by the rate, t = ln(2)/r.
Plugging in the values from the question, we have r = 0.04 (4% interest rate) and thus t = ln(2)/0.04.
When we calculate this, we get t ≈ 17.33 years.
Therefore, an investment of $2000 at a 4% annual interest rate compounded continuously will take approximately 17.33 years to double.