Final answer:
To determine how long it takes for an investment of $9,000 at 9% APR compounded monthly to grow to $27,000, one must solve the compound interest formula for 't'. After applying the given numbers and mathematical operations, the investment will reach $27,000 in approximately 4 years, leading to the correct option b) 4 years.
Step-by-step explanation:
The student's question involves determining how long it will take for an investment to triple in value under a specific compound interest formula.
We're given the initial investment amount of $9,000, an annual percentage rate (APR) of 9% that is compounded monthly, and the future value formula S = 9,000(1 + 0.09/12)^(12t), where 'S' is the future value and 't' is the time in years. To find out how many years it takes for the investment to reach $27,000, we apply the values to the formula and solve for 't'.
To do this, substitute $27,000 for 'S' in the formula and solve for 't':
27,000 = 9,000(1 + 0.09/12)^(12t).
Dividing both sides by 9,000 gives us 3 = (1 + 0.09/12)^(12t).
Taking the natural logarithm of both sides gives us ln(3) = 12t * ln(1 + 0.09/12).
Solving for 't' gives us approximately 4 years. Therefore, the answer is b) 4 years.