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Geometry

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Zoe the goat is tied by a rope to one corner of a 15 meter-by-25 meter rectangular barn in the middle of a large grassy field. Over what area of the field can Zoe graze if the rope is 10 meters long? 20 meters long? 30 meters long?

  • Geometry -

    a 10m rope allows grazing over 3/4 of a 10m circle

    3/4 * π * 100 = 75π

    a 20m rope allows grazing over 3/4 of a 20m circle + 1/4 of a 5m circle

    3/4 * π * 400 + 1/4 * π * 25 = 306.25 π

    a 30m rope allows
    3/4 of a 30m circle
    + 1/4 of a 15m circle
    + 1/4 of a 5m circle

    you asked where I got the 5.

    the 30m rope wraps around the corner at the other end of the 25m side, leaving a 5m section of rope to extend along the short side there.

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