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Date May 2016 Marks available 6 Reference code 16M.2.hl.TZ1.4
Level HL only Paper 2 Time zone TZ1
Command term Find Question number 4 Adapted from N/A

Question

The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The points A and C are 3 m apart. A goat GG is tied by a 5 m length of rope at point A on the outside edge of the enclosure.

Given that the corner of the enclosure at C forms an angle of θθ radians and the area of field that can be reached by the goat is 44 m244 m2, find the value of θθ.

M16/5/MATHL/HP2/ENG/TZ1/04

Markscheme

attempting to use the area of sector formula (including for a semicircle)     M1

semi-circle 12π×52=25π2=39.2699081712π×52=25π2=39.26990817     (A1)

angle in smaller sector is πθπθ     (A1)

area of sector =12×22×(πθ)=12×22×(πθ)     (A1)

attempt to total a sum of areas of regions to 44     (M1)

2(πθ)=4439.269908172(πθ)=4439.26990817

θ=0.777 (=29π422)θ=0.777 (=29π422)     A1

Note:     Award all marks except the final A1 for correct working in degrees.

Note:     Attempt to solve with goat inside triangle should lead to nonsense answer and so should only receive a maximum of the two M marks.

[6 marks]

Examiners report

Many students experienced difficulties with this question, mostly it seems through failing to understand the question. Some students left their answers in degrees, thereby losing the final mark.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.1 » The circle: radian measure of angles.
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