Two ladders--9 meters and 6 meters high each--are set up in an alley such that one ladder leans from the base of the left wall to the right wall, and the second ladder leans from the base of the right wall to the left wall, as shown. The two ladders cross exactly 3 meters above the ground.
Determine the width of the alleyway in meters.
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⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ tan α = x 3 tan β = a − x 3 ⟹ tan ( cos − 1 9 a ) = x 3 ⟹ tan ( cos − 1 6 a ) = a − x 3 ⟹ a 8 1 − a 2 = x 3 ⟹ a 3 6 − a 2 = a − x 3 ⟹ x = 8 1 − a 2 3 a ⟹ x = a − 3 6 − a 2 3 a
Therefore,
8 1 − a 2 3 a 8 1 − a 2 3 = a − 3 6 − a 2 3 a = 1 − 3 6 − a 2 3 Since a = 0
Solving the equation numerically, we have a ≈ 3 . 6 9 4 .