A fence 2 feet tall runs parallel to a tall building at a distance of 3 feet from the building.?
A fence 2 feet tall runs parallel to a tall building at a distance of 3 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
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It’s a geometry problem… draw a diagram and use similar triangles.
)A fence 3 feet tall runs parallel to a tall building at a distance of 2 feet from the building. What is the length of the shortest ladder
by drawing a diagram and labeling it with x being the distance from the fence to the bottom of the ladder, Z being the distance from the top of the ladder and bottom of the tall biulding and Y being the length of the ladder you will get 2 eqations Z^2 + (3+X)^2 = Y^2 (pythagorus theorem) and by similar triangles (3+X)/X = Z/2 ===> this gives Z= 2*((3+X)/X) substitute this into the first eqation and you get
(2*((3+X)/X))^2 + (3+X)^2 = Y^2
=sqrt (2*((3+X)/X))^2 + (3+X)^2)
graph this and there will be a minimum at X=2.289 and Y = 7.02 with Y being the min length of the ladder and X being the distance from the fence to the bottom of the ladder, with this you can find Z but it isnt needed
It’s wise to draw a picture and label it with the given data:
fence height: 2 ft
distance from ladder point of contact with ground
to base of bldgs’ wall: 3 ft.
angle of the ladder θ with the ground(to be determined)
And now, you may calculate the length of ladder as follows:
Observe from your drawing that;
L = (3 + d’) / cosθ
where,
Distance between the fence and the ladder’s point of contact with the ground: d’
Angle the ladder makes to the horizontal: θ
d’ = 2/tanθ,
L = (3 + 2/tanθ) / cosθ = 3/cosθ + 2/sinθ
Differentiate L and set equal to zero to find the
minimum value of L:
dL/dθ = 3tanθ/cosθ – 2/(tanθ*sinθ = 0
3tan²θ – 2/tanθ = 0
(3/2)tan³θ – 1 = 0
(3/2) tan³θ = 1
tan³θ = 2/3
tan θ = ³√2/3
θ = 41 deg.
Plug the value of θ = 41 deg into eqn. for L
derived above:
L = 3/cosθ + 2/(sinθ
L = 3/cos(41) + 2/sin(41) = 3.98 + 3.05 = 7.03 ft