Finding the time in a word problem on compound interest
Latoya needs $\$ 4740$ for a future project. She can invest $\$ 4000$ now at an annual rate of $3 \%$, compounded monthly. Assuming that no withdrawals are made, how long will it take for her to have enough money for her project?

Do not round any intermediate computations, and round your answer to the nearest hundredth.
Dyears
$x \quad 0$

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.