11. Two farmers have bought some farmland between them. The farmland is in the shape of a quadrilateral ABCD, as shown below.
Diagram not drawn to scale
The farmers want to divide the farmland equally by building a straight fence.
(a) One of the farmers has suggested building the fence from A to C. Show that this does not divide the farmland equally. [3]
(b) To divide the land equally, the fence is built from A to F where CF = 17·9 m. To construct the fence, posts are placed at A and F. Other posts are then located along AF, so that the posts are no more than 3 m apart.
Calculate the smallest number of posts needed, including the posts at A and F. [5]
Diagram not drawn to scale
From the formula list: In any triangle \(ABC\): Cosine rule \(a^2 = b^2 + c^2 - 2bc\cos A\); Area of triangle \(= \dfrac{1}{2}ab\sin C\)
Allow use of 18 instead of 17.9 AND/OR 95 instead of 94.9 for M2 only M1 for (AF2 =) 1642 + (190–17.9)2 – 2×164×(190–17.9)×cos94.9(º) (AF2 = 61336.09…) OR M1 for use of 190 instead of (190–17.9)
= 247.6(…) or 247.7 or 248 (m)
A1
CAO. Can only be awarded if (190–17.9 =) 172.1 and 94.9 used in the cosine rule
84 (posts needed)
B1
FT ‘their derived 247.6(…)’ ÷ 3 (rounded up) + 1 provided rounding required OR FT ‘their derived 247.6(…)’ ÷ 3 (rounded down) + 2 provided rounding required
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