Final answer:
Jack will have to wait approximately 23.45 years for his $15,000 deposit to grow to $29,000 with a 3% annual interest rate compounded monthly.
Step-by-step explanation:
Jack needs to wait approximately 23.45 years for his investment to grow to $29,000 at a 3% annual interest rate compounded monthly.
Starting with the formula for compound interest P(1+ \frac{r}{n})^{nt} = A, where P is the principal amount, r is the annual interest rate (in decimal), n is the number of times interest is compounded per year, t is the time the money is invested for in years, and A is the amount of money accumulated after n years, including interest.
Given: P = $15,000, r = 0.03, n = 12 (compounded monthly), and A = $29,000. Rearrange the formula to solve for t:
\frac{A}{P} = (1+ \frac{r}{n})^{nt} => \frac{29,000}{15,000} = (1+\frac{0.03}{12})^{12t}
Calculate the left side of the equation to get:
1.9333... = (1+0.0025)^{12t}
Now, we take the natural logarithm of both sides to solve for t:
ln(1.9333...) = ln((1+0.0025)^{12t}) => ln(1.9333...) = 12t * ln(1+0.0025)
Solving for t gives us:
t = \frac{ln(1.9333...)}{12 * ln(1+0.0025)} \approx 23.45