MY 11 TRISECTION APPROXIMATION METHODS Pg. 36
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FIG. 6
FIG. 7
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An alleyway has a 40 feet extension ladder leaning with its foot
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against one wall and its top end against the opposite wall, while just
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touching an overhang 18 feet above the pavement,
as shown in FIG. 6
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above. Another 20 foot ladder similarly
leans against opposite walls
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and crosses the extension ladder 6 feet above the lane, see FIG. 6
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| Pg. 37
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and FIG. 7 above.
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What is the width of the alleyway?
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In FIG. 7:
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Since H/C = 18/6 = 3 & R >= H
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therefore ÐAOB = 3Æ
and ÐAOC = Æ
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ie. my [KLAUSIAN TRI-FOCUS LAW]
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Therefore sin3Æ = 18/20 = 0.9
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3Æ = 64.15806o
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Æ = 21.38602o
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Thus the alley width W = 20cosÆ = 18.6229 feet = 18' 7 15/32''
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Z = 20cos3Æ = (R2 + H2)½ = (202 + 182)½ = 8.7178feet
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Ladder length = (X2 + W2)½ = 35.94047 feet
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X/W = 18/(W - Z) = tanÐOAD
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X = 18W/(W-Z) = 18*18.6229/(18.6229 - Z) = 30.739 feet.
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X = 18(18.6229)/9.905098 = 30.739 feet.
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Using the focus formula: 1/C = 1/X + 1/Y
[from CRC TABLES]
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1/Y = 1/C - 1/X = 1/6 - 1/30.739
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Y = 7.292993 feet.
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Y = 20sinÆ = 20sin21.386o = 7.292993 feet
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KLAUSIAN TRI-FOCUS COSINE LAW: C = H/3 = 20sin3Æ/3 = 20(.9)/3 = 6
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Using the focus formula: 1/C = 1/X + 1/Y
[from CRC TABLES]
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X = C*Y/(Y - C) = 6(7.292993)/(7.292993 - 6) = 30.739 feet.
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| Pg. 39
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H = 3C = 3(6) = 18 feet.
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X/W = H/(W - Z)
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(W - Z) = H*W/X
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Z = W - H*W/X = 18.6229 - 18(18.6229)/33.842 = 8.71780 feet
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[from CRC TABLES]:
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sin3Æ/sinÆ - cos3Æ/cosÆ = (3 - 4sin2Æ) - (1 - 4sin2Æ)
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sin3Æ/sinÆ - cos3Æ/cosÆ = 2
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ie. my [KLAUSIAN TRI-FOCUS COSINE LAW]
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