What is the length of the shortest ladder that will reach from the ground over the fence, 3 feet tall which runs parallel to a tall building at a distance of 2 feet from the bulding?
Answer 280.
Refr to the figure as under.
Let l = length of the ladder
and θ = its inclination with the ground towards the wall.
=> lsinθ / 3 = lcosθ / (lcosθ - 2)
... ... ... ... [Using the property of similar triangles]
=> sinθ (lcosθ - 2) = 3cosθ
=> l = 3cosecθ + 2secθ
For l to be minimum,
dl/dθ = 0 and d^2l/dθ^2 > 0
dl/dθ = 0
=> - 3cosecθ cotθ + 2secθ tanθ = 0
=> tan^3 θ = 3/2
=> θ = arctan (3/2)^(1/3)
=> θ = 48.86°
=> l = 3cosec(48.86°) + 2sec(48.86°)
=> l = 3.9835 + 3.0400
=> l = 7.0235 feet.
d^2l/dθ^2 = 3cosecθ cot^2 θ + 3cosec^3 θ + 2secθ tan^2θ + 2sec^3 θ > 0 for acute θ thus proving that length of ladder, l, found is minimum.
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