Can someone help with this trig problem?

Two fire towers are 40 miles apart, tower A being due west of tower B. A fire is spotted from the towers, and the angle of elevations from A and B are 17degrees and 38degrees, respectively. Find the distance d of the fire from the line segment AB.

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  • 7 years ago
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    Let's call the fire location F, and the closest point on AB to the fire G.

    We're looking for distance d = FG. It's an altitude of triangle ABF, which means that AGF and BGF are both right angles.

    Let's call the distance AG = x.

    Then BG = 40-x

    tan(17) = d/x

    x * 0.3057 = d

    tan(38) = d/(40-x)

    (40-x) * 0.7813 = d

    Substitute for d:

    x * 0.3057 = (40-x) * 0.7813

    x * 1.087 = 31.252

    x = 28.75

    Then put that x back into either of the original equations to get d = 8.789.

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