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In order to build a bridge across a river, a surveyor is hired to find the distance across the river. The surveyor places a stake on either side of the river. As shown in the diagram below (the diagram is not drawn to scale). He measures the bearing to the first stake as 16 degrees East of North at a distance of 562 feet. He measures the departure of the second stake as 330.87 feet West and the latitude as 198.80 feet North. Find the distance across the river. Hint: label the diagram and use the information given to help form a triangle then apply one of the laws to solve for the missing side

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asked by Anonymous
  1. Interpretation of given information:
    The location of the theodolite (surveyor) at point A.
    First stake (Point B)
    Second stake (Point C).
    Points B and C are on opposite sides of the river, and distance BC represents the width of the river.
    For point B:
    distance = 562
    bearing = 16° (from north clockwise)
    For point C:
    departure (Easting) = 330.87'
    Latitude (northing) 198.8'
    equivalent to:
    distance = √(330.87²+198.8²)
    =386.0005
    bearing=90-atan(198.8/330.87)=61.6135°

    Let's form triangle ABC,
    AB=562'
    AC=386.0005'
    ∠BAC=61.6135-16=45.6135°

    Use cosine rule to find distance BC
    BC²=562²+386.0005²-2(562)(386.0005)cos(45.6135°)
    =161354.304
    BC=√(161354.304)=401.7'

    Width of bridge = 401.7'

    (Do check the calculations for accuracy because I used the Google calculator with which I am not familiar.)

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    posted by MathMate

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