Final answer:
Using the formula for continuous compound interest, we find that $14,000 needs to be invested for approximately 7.16 years at an 11% interest rate compounded continuously to earn $13,087.09 in interest.
Step-by-step explanation:
To calculate how long $14,000 must be invested at 11% interest compounded continuously to earn $13,087.09 in interest, we can 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 sum of money), r is the annual interest rate (decimal), t is the time the money is invested in years, and e is Euler's number (approximately 2.71828).
First, we need to calculate the total amount A after the investment period, which is the initial investment plus the interest earned: A = $14,000 + $13,087.09 = $27,087.09.
Now we can solve for t using the formula:
27,087.09 = 14,000 * e(0.11*t)
To isolate t, we can take the natural logarithm (ln) of both sides:
ln(27,087.09/14,000) = ln(e(0.11*t))
ln(27,087.09/14,000) = 0.11*t
t = ln(27,087.09/14,000) / 0.11
After calculating, we find that t is approximately 7.16 years.