Trigonometry

A vertical pole 35m high, standing on sloping ground is braced by a wire which extends from the top of the pole to a point on the ground 25 m from the foot of the pole. If the pole subtends an angle of 30 degrees at the point where the wire reaches the ground, how long is the wire?

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  1. as usual, draw a diagram. Let
    T be the top of the pole
    B be the bottom of the pole
    P be the point where the wire meets the ground.

    Draw a horizontal line from P, and extend TB into the ground, so that
    Q is the intersection of the horizontal from P and the vertical from T

    Then if we let
    x be the horizontal distance PQ
    y be the vertical distance QB
    z be the length of the wire PT
    θ be the angle the road makes with the horizontal
    we have

    y/x = tanθ
    (y+35)/x = tan(θ+30°)
    So,

    tan(θ+30°) = (tanθ + 1/√3)/(1 - tanθ/√3)
    (y/x + 1/√3)/(1 - y/x√3) = (x+35)/y
    x^2+y^2 = 25^2
    That gives us
    x = 18.44 and y=16.88

    z^2 = x^2 + (y+35)^2
    z^2 = 18.44^2 + (16.88+35)^2
    z = 55.06

    So, the wire is about 55m long.

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  2. My bad - switched x and y:

    (y/x + 1/√3)/(1 - y/x√3) = (y+35)/x
    x^2+y^2 = 25^2
    x = 19.408, y=15.759
    so, z = 54.343

    not very far off, really, eh?

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  3. it s laborious using a right triangle theorem, however if sine and cosine law for oblique triangle were discussed by the instructor, it s an easy
    work.

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