Final answer:
To solve for the time required to grow $11,000 to $19,000 at a 3% annual interest rate compounded monthly, utilize the compound interest formula. Rearrange the formula to solve for time, t, and insert the known values. Calculate and round to two decimal places.
Step-by-step explanation:
To determine how long it will take for $11,000 to grow to $19,000 at an interest rate of 3% per year compounded monthly, we can use the formula for compound interest which is given by:
A = P(1 + r/n)nt
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).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
In this case, A = $19,000, P = $11,000, r = 0.03 (3%), and n = 12 (since the interest is compounded monthly). We are solving for t, the time in years.
We can rearrange the formula to solve for t:
t = (ln(A/P)) / (n * ln(1 + r/n))
Now plug in the values:
t = (ln(19,000 / 11,000)) / (12 * ln(1 + 0.03/12))
Calculate the values in the logarithm and perform the division to find t.
You would round your answer to two decimal places to get the final answer.