How much time will be needed for $\$ 34,000$ to grow to $\$ 37,557.15$ if deposited at $4 \%$ compounded quarterly? The amount $\$ 34,000$ will grow to $\$ 37,557.15$ in year(s). (Do not round until the final answer. Then round to the nearest tenth as needed.)

Step 1 ：We are given the following values: Principal amount (P) = $34,000, Final amount (A) = $37,557.15, Annual interest rate (r) = 4% or 0.04, and the number of times the interest is compounded per year (n) = 4.

Step 2 ：We need to find the time in years (t) for the principal amount to grow to the final amount. We can use the formula for compound interest, rearranged to solve for t: \(t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)}\).

Step 3 ：Substituting the given values into the formula, we get \(t = \frac{\log(37557.15/34000)}{4 \cdot \log(1 + 0.04/4)}\).

Step 4 ：Solving the above expression, we get t = 2.4999984854534882.

Step 5 ：Rounding to the nearest tenth, we get t = 2.5.

Step 6 ：Final Answer: The amount $34,000 will grow to $37,557.15 in approximately \(\boxed{2.5}\) years.