Math

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Two fire towers are 30 kilometers apart, tower A being due west of tower B. A fire is spotted from the towers, and the bearings from A and B are E 14 degrees N and W 34 degrees N, respectively. Find the distance d of the fire from the line segment AB.

  • Math -

    1. Draw line segment AB.

    2. Draw the 14-degree angle from point
    A.

    3. Draw the 34-degree angle from point B. Label the intersection of these 2 lines point C. Now we have formed
    triangle ABC.

    4. Draw the altitude from point C perpendicular to AB and label it CD.

    The distance of the fire from AB is
    equal to the altitude(CD).

    A + B + C = 180 Deg.
    14 + 34 + C = 180,
    C = 132 Deg.


    a/sinA = c/sinC,
    a/sin14 = 30/sin132,
    Multiply both sides by sin14:
    a = 30sin14 / sin132 = 9.77km.

    CD = 9.77sin34 = 5.46km = dist. from
    fire to AB.

  • Math -

    21.9

  • Math -

    1. Call the point where d intersects AB point C.
    2. Let CB equal x.
    3. cot(14)= (30-x)/d
    cot(34)= x/d
    4. cot(14)= (30/d)- (x/d)
    cot(14)= (30/d)- cot(34)
    cot(14)+ cot(34)= (30/d)
    d(cot14+ cot34)= 30
    d = 30/ (cot14+ cot34)
    d = 5.46 km

  • Math -

    Sorry this one's easier to read.
    1..
    Call the point where d intersects AB point C.

    2..
    Let CB equal x.

    3..
    cot(14)= (30-x)/d
    cot(34)= x/d

    4..
    cot(14)= (30/d)- (x/d)
    cot(14)= (30/d)- cot(34)
    cot(14)+ cot(34)= (30/d)
    d(cot14+ cot34)= 30
    d = 30/ (cot14+ cot34)
    d = 5.46 km

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