How much time will be needed for $\$ 17,000$ to grow to $\$ 17,779.49$ if deposited at $3 \%$ compounded quarterly?

Step 1 ：We are given that the principal amount (P) is $17,000, the final amount (A) is $17,779.49, the annual interest rate (r) is 3% or 0.03 in decimal form, and the interest is compounded quarterly, so n = 4.

Step 2 ：We can use the formula for compound interest, which is A = P (1 + r/n)^(nt), and rearrange it to solve for t: t = log(A/P) / (n * log(1 + r/n)).

Step 3 ：Substituting the given values into the formula, we get t = log(17779.49/17000) / (4 * log(1 + 0.03/4)).

Step 4 ：Calculating the above expression, we find that t is approximately 1.5000037697808604.

Step 5 ：Rounding to the nearest tenth, we find that the time needed for $17,000 to grow to $17,779.49 if deposited at 3% compounded quarterly is approximately \(\boxed{1.5}\) years.