MY 11 TRISECTION APPROXIMATION METHODS         Pg. 36

 

THE ALLEYWAY PROBLEM

 
tri6c.gif


FIG. 6

tri7c.gif
FIG. 7

 

PROBLEM:

 
An alleyway has a 40 feet extension ladder leaning with its foot
 
against one wall and its top end against the opposite wall, while just
 
touching an overhang 18 feet above the pavement, as shown in FIG. 6
 
above. Another 20 foot ladder similarly leans against opposite walls
 
and crosses the extension ladder 6 feet above the lane, see FIG. 6
  Pg. 37
and FIG. 7 above.
 
What is the width of the alleyway?
 

SOLUTION:

 
In FIG. 7:
 
Since H/C = 18/6 = 3 & R >= H
 
therefore ÐAOB = 3Æ and ÐAOC = Æ
 
ie. my [KLAUSIAN TRI-FOCUS LAW]
 
Therefore sin3Æ = 18/20 = 0.9
 
3Æ = 64.15806o
 
Æ = 21.38602o
 
Thus the alley width W = 20cosÆ = 18.6229 feet = 18' 7 15/32''
  Pg. 38

DIMENSIONS:

 
Z = 20cos3Æ = (R2 + H2)½ = (202 + 182)½ = 8.7178feet
 
Ladder length = (X2 + W2)½ = 35.94047 feet
 
X/W = 18/(W - Z) = tanÐOAD
 
X = 18W/(W-Z) = 18*18.6229/(18.6229 - Z) = 30.739 feet.
 
X = 18(18.6229)/9.905098 = 30.739 feet.
 
Using the focus formula: 1/C = 1/X + 1/Y   [from CRC TABLES]
 
1/Y = 1/C - 1/X = 1/6 - 1/30.739
 
Y = 7.292993 feet.
 

CHECK:

 
Y = 20sinÆ = 20sin21.386o = 7.292993 feet
 
KLAUSIAN TRI-FOCUS COSINE LAW: C = H/3 = 20sin3Æ/3 = 20(.9)/3 = 6
 
Using the focus formula: 1/C = 1/X + 1/Y   [from CRC TABLES]
 
X = C*Y/(Y - C) = 6(7.292993)/(7.292993 - 6) = 30.739 feet.
  Pg. 39
H = 3C = 3(6) = 18 feet.
 
X/W = H/(W - Z)
 
(W - Z) = H*W/X
 
Z = W - H*W/X = 18.6229 - 18(18.6229)/33.842 = 8.71780 feet
 
[from CRC TABLES]:
 
sin3Æ/sinÆ - cos3Æ/cosÆ = (3 - 4sin2Æ) - (1 - 4sin2Æ)
 
sin3Æ/sinÆ - cos3Æ/cosÆ = 2
 
ie. my [KLAUSIAN TRI-FOCUS COSINE LAW]
 
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