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urgent maths pls

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A goat is tied at point A with a rope of length(L). How long can L be so that the goat can feed on exactly half of the field if the field is a circle.

  • urgent maths pls -

    Assuming A is on the edge of the field (circle F), let the radius of the field be R, and the rope be length L.

    Let the circle with radius L and center A intersect the field at points B and C.

    Then we want the area enclosed by the arcs BC to be 1/2 pi R^2.

    The sector of circle A subtended by arc BC has area 1/2 L^2 θ where θ subtends BC.

    The rest of the area comprises two segments of circle F, where each has area

    1/2 R^2 (φ - sinφ)

    where φ subtends arc AB or AC.

    So, we want


    1/2 L^2 θ + R^2 (φ - sinφ) = pi/2 R^2

    So, what are θ and φ?

    In triangle FAB, let FB be s.

    s^2 = L^2 + R^2 - 2RL cos θ/2
    s/sin θ/2 = L/sinφ

    That should yield the result you need.

  • urgent maths pls -

    Pls d answer isn't really clear. And where did d triangle come from??

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