Answer:
2.436 ft . . . or . . . 2 ft 5.228 in.
Explanation:
The middle length side of a 30°-60°-90° right triangle is √(3/4) times the length of the longest side. That means the bracing pieces have lengths that follow a geometric sequence with initial value 5√(3/4) and common ratio √(3/4). We want to find the value of n such that ...
an < 2.500
5√(3/4)·(√(3/4))^(n-1) < 2.500
Taking logarithms, we have ...
log(5√(3/4)) + (n-1)·log(√(3/4)) < log(2.500)
(n-1)·log(√(3/4)) < log(2.500) -log(5√(3/4))
Now, the log of √(3/4) is negative, so when we divide by that value, we must reverse the inequality symbol.
n -1 > (log(2.500) -log(5√(3/4)))/log(√(3/4))
n > 1 + 3.8
That is, the 5th brace (GH) will be shorter than 2.5 feet. Its length will be ...
5√(3/4)·(√(3/4))^(5-1) = (45/16)√(3/4) ≈ 2.43570 ft
In feet and inches, that is 2 feet + 12·0.43570 in ≈ 2 ft 5.2284 in.
Correct to 3 decimal places, the longest brace shorter than 2.5 ft is brace GH at 2.436 ft, or 2 ft 5.228 inches.
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Comment on decimal places
The problem numbers are given both in feet and in feet-and-inches. When it asks for 3 decimal places, it is not clear whether that's 3 decimal places of feet, or 3 decimal places of inches.
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Alternate solution method
You know the number of braces will not be terribly large, so you can just figure the length of each one:
AD = 0.8660254·AB ≈ 4.330127
DE = 0.8660254·AD = 3.75 (exact)
EF = 0.8660254·DE ≈ 3.247595
FG = 0.8660254·EF = 2.8125 (exact)
GH = 0.8660254·FG ≈ 2.435696 . . . . . this is the one we're looking for