Step 1 :Given that Latoya needs $4740 for a future project and she can invest $4000 now at an annual rate of 3%, compounded monthly. We are to find out how long it will take for her to have enough money for her project.
Step 2 :We can use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\), 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 (in decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.
Step 3 :We can rearrange the formula to solve for t: \(t = \frac{\ln(\frac{A}{P})}{n \ln(1 + \frac{r}{n})}\)
Step 4 :Substituting the given values into the formula, we get \(t = \frac{\ln(\frac{4740}{4000})}{12 \ln(1 + \frac{0.03}{12})}\)
Step 5 :Calculating the above expression, we find that \(t \approx 5.67\)
Step 6 :So, it will take approximately \(\boxed{5.67}\) years for Latoya to have enough money for her project.