Date | November 2021 | Marks available | 6 | Reference code | 21N.1.AHL.TZ0.11 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 0 |
Command term | Find | Question number | 11 | Adapted from | N/A |
Question
The following diagram shows a corner of a field bounded by two walls defined by lines L1 and L2. The walls meet at a point A, making an angle of 40°.
Farmer Nate has 7 m of fencing to make a triangular enclosure for his sheep. One end of the fence is positioned at a point B on L2, 10 m from A. The other end of the fence will be positioned at some point C on L1, as shown on the diagram.
He wants the enclosure to take up as little of the current field as possible.
Find the minimum possible area of the triangular enclosure ABC.
Markscheme
METHOD 1
attempt to find AC using cosine rule M1
72=102+AC2-2×10×AC×cos 40° (A1)
attempt to solve a quadratic equation (M1)
AC=4.888… AND 10.432… (A1)
Note: At least AC=4.888… must be seen, or implied by subsequent working.
minimum area =12×10×4.888…×sin(40°) M1
Note: Do not award M1 if incorrect value for minimizing the area has been chosen.
=15.7 m2 A1
METHOD 2
attempt to find AˆCB using the sine Rule M1
sin C10=sin 407 (A1)
C=66.674…° OR 113.325…° (A1)
EITHER
B=180-40-113.325…
B=26.675…° (A1)
area =12×10×7×sin(26.675…°) M1
OR
sine rule or cosine rule to find AC=4.888… (A1)
minimum area =12×10×4.888…×sin(40°) M1
THEN
=15.7 m2 A1
Note: Award A0M1A0 if the wrong length AC or the wrong angle B selected but used correctly finding a value of 33.5 m2 for the area.
[6 marks]
Examiners report
As has often been the case in the past, trigonometry is a topic that is poorly understood and candidates are poorly prepared for. Approaches to this question required the use of the cosine or sine rules. Some candidates tried to use right-angled trigonometry instead. A minority of candidates used the cosine rule approach and were more likely to be successful, navigating the roots of the quadratic equation formed. When using the sine rule the method involved the ambiguous case as the required angle was obtuse. Few candidates realized this and this was the most common mistake. In a few instances, the word “minimum” led candidates to attempt an approach using calculus.