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A lighthouse keeper 100 feet above the water sees a boat sailing in a straight line directly toward her. As watches, the angle of depression to the boat changes from 25 degrees to 40. How far has the boat traveled during this time?

  • Pre-Calculus-Trig -

    First, we draw a rectangle and label
    the ver. sides 100ft and hor.(longer)
    sides X ft. Then we draw a diag. from
    upper left to lower rt. side forming a rt triangle. The angle of elevation = the angle of depression = 25deg.
    Draw a 2nd line from upper left corner
    to some point(near the center) on the
    bottom hor. line forming an angle of
    40deg with the hor. Label the 2 sections on the bottom line X1 and X2
    (rt to lt).

    tan25 = Y/X = 100/X,
    X = 100 / tan25 = 214.45 ft.

    tan40 = Y/X2 = 100/X2,
    X2 = 100 / tan40 = 119.2ft.

    X1 + X2 = X,
    X1 + 119.2 = 214.5,
    X1 = 214.5 - 119.2 = 95.3ft. = distance traveled.

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