On February 1, 1977, John deposited $2250 into a savings account paying 5.76% interest, compounded quarterly. If he hasn't made any additional deposits or withdrawals since then, and if the interest rate has stayed the same, in what year did his balance hit $4500, according to the rule of 72?

John's effective annual rate is about
(1 +.0576/4)^4 -1 ≈ 5.8856%
According to the "rule of 72", John's money will have doubled in
72/5.8856 = 12.23 years

John's balance will be $4500 in 1989.

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Since you're only concerned with the year (not the month), you don't actually need to determine the effective annual rate. The given rate of 5.76% will tell you 72/5.76 = 12.5 years. The actual doubling time is closer to 12.12 years, so using the effective rate gives results that are closer, but "good enough" is good enough in this case.