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Two Boats left the harbour at the same time. One travelled at 10km/h on a bearing of N47°E. The other travelled at 8km/h on a bearing of N79°E. How far apart were the boats after 78 mins? Round the distance to the nearest tenth of a kilometer. Use trigonometry.

  • Math -

    To understand this you really need to sketch this out.

    Think of the x/y axis on a graph, with the y-axis as North and the x-axis as East. Both boats are traveling from the North (top of the y-axis), toward East (down toward the x-axis).

    The first bearing, forms a 47 deg. angle with the y-axis and the second bearing, goes a little further forming a 79 deg. angle with the y-axis.

    The triangle formed, (I labeled ABC), with A at the origin, side c (between angle A and angle B), side b (between angle A and angle C) and side a (between angle B and angle C).

    You need to find side a, the distance between the boats.

    1 km/h / 60 min = 0.01667 km/min
    First boat 10 km/h * 0.01667 = 0.1667 km/min * 78 min = 13 km
    Second boat 8 km/h * 0.01667 = 0.1333 km/min * 78 min = 10.4 km

    Angle A = 79 deg - 47 deg = 32 deg.
    side c = 13 km (first boat's bearing)
    side b = 10.4 km (second boat's bearing)

    To find side a,
    a^2 = b^2 + c^2 - 2bc cos A
    a^2 = 10.4^2 + 13^2 - 2(10.4)(13)cos 32d
    a^2 = 108.16 + 169 - 270.4 (.8)
    a^2 = 277.16 - 216.32
    a^2 = 60.8 km

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