Question 3 of 17. Step 1 of 1

Donnie is saving up money for a down payment on a motorcycle. He currently has $\$ 4363$, but knows he can get a loan at a lower interest rate if he can put down $\$ 5121$, If he invests the $\$ 4363$ in an account that earns $5.2 \%$ annually, compounded monthly, how long will it take Donnie to accumulate the $\$ 5121$ ? Round your answer to two decimal places, If necessary.

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Step 1 ：First, we need to convert the interest rate from percentage to decimal by dividing it by 100. The principal amount \(P\) is \$4363, the amount \(A\) is \$5121, the interest rate \(r\) is 5.2% or 0.052, and the number of times interest is compounded per year \(n\) is 12 (since it's compounded monthly).

Step 2 ：We can plug these values into the formula for compound interest and calculate \(t\). The formula is \(t = \frac{\ln(\frac{A}{P})}{n \cdot \ln(1 + \frac{r}{n})}\).

Step 3 ：Substituting the given values into the formula, we get \(t = \frac{\ln(\frac{5121}{4363})}{12 \cdot \ln(1 + \frac{0.052}{12})}\).

Step 4 ：Calculating the above expression, we get \(t \approx 3.09\).

Step 5 ：So, it will take Donnie approximately 3.09 years to accumulate the \$5121.

Step 6 ：Final Answer: \(\boxed{3.09}\)