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calculus

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A straight road makes an angle, A of 12 degrees. When the angle of elevation, B, of the sun is 55 degrees, a vertical pole beside the road casts a shadow 7 feet long parallel to the road. Approximate the length of the pole. Round to tow decimal places.

  • calculus - ,

    draw a diagram.
    P = end of shadow on road
    T = top of pole
    Q = foot of pole
    Z = intersection of horizontal line from Q and pole extended vertically.

    Let d = PT: the distance from the tip of the shadow to the top of the pole
    Let x = PZ
    Let y = ZQ
    Let h = QT, the height of the pole

    We know:
    ∠ZPT = 55°
    ∠ZPQ = 12°
    so, ∠QPT = 43°
    PQ = 7

    x = 7cos12° = 6.847
    y = 7sin12° = 1.455
    d = x/cos55° = 7cos12°/cos55° = 11.937

    now, we can do this two ways
    (1) Pythagorean Theorem
    x^2 + (y+h)^2 = d^2
    h = 8.323

    (2) Law of Cosines
    h^2 = d^2 + 7^2 - 2*7*d*cos43°
    h = 8.323

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