Answer giving many ! the diagram on the right shoes a metal component shaped like a right triangle with a number of internal bracing pieces that meet alternate sides of the 30° angle at right angles. correct to 3 decimal places, what is the length of the longest branching price that's shorter than 2 ft 6 in?

  2.436 ft . . . or . . . 2 ft 5.228 in.

Step-by-step 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.

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

i dont know i just need points

huh? im pretty sure it is in 199

step-by-step explanation: