Date | May 2022 | Marks available | 5 | Reference code | 22M.2.SL.TZ2.7 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 2 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
A farmer is placing posts at points A, B, and C in the ground to mark the boundaries of a triangular piece of land on his property.
From point A, he walks due west 230 metres to point B.
From point B, he walks 175 metres on a bearing of 063° to reach point C.
This is shown in the following diagram.
The farmer wants to divide the piece of land into two sections. He will put a post at point D, which is between A and C. He wants the boundary BD to divide the piece of land such that the sections have equal area. This is shown in the following diagram.
Find the distance from point A to point C.
Find the area of this piece of land.
Find CÂB.
Find the distance from point B to point D.
Markscheme
AˆBC=27° (A1)
attempt to substitute into cosine rule (M1)
1752+2302-2(175)(230)cos 27° (A1)
108.62308…
AC=109 (m) A1
[4 marks]
correct substitution into area formula (A1)
12×175×230×sin 27°
9136.55…
area =9140 (m2) A1
[2 marks]
attempt to substitute into sine rule or cosine rule (M1)
sin 27°108.623…=sin ˆA175 OR cos A=(108.623…)2+2302-17522×108.623…×230 (A1)
47.0049…
CÂB=47.0° A1
[3 marks]
METHOD 1
recognizing that for areas to be equal, AD=DC (M1)
AD=12AC=54.3115… A1
attempt to substitute into cosine rule to find BD (M1)
correct substitution into cosine rule (A1)
BD2=2302+54.31152-2(230)(54.3115)cos 47.0049°
BD=197.009…
BD=197 (m) A1
METHOD 2
correct expressions for areas of triangle BDA and triangle BCD using BD A1
12×BD×230×sin x° and 12×BD×175×sin (27-x)° OR
12×BD×230×sin (27-x)° and 12×BD×175×sin x°
correct equation in terms of x (A1)
175 sin(27-x)=230 sin x or 175 sin x=230 sin(27-x)
x=11.6326… or x=15.3673… (A1)
substituting their value of x into equation to solve for BD (M1)
12×BD×230×sin 11.6326…=12×BD×175×sin 15.3673… or
12×BD×230×sin 11.6326…=12×9136.55…
BD=197 (m) A1
[5 marks]
Examiners report
Students performed well on parts (a)-(c), correctly applying the cosine rule, the sine formula for area and the sine rule. Part (d) proved challenging. A common error was to falsely assume that segment BD bisected angle ABC.
A significant number of candidates did not have their calculator in degree mode or started in radians and changed to degrees part way through but used answers they had obtained when they were in radian mode. They got answers which were clearly impossible from the diagram, but most did not notice this.
Accuracy was a great problem throughout this question: premature rounding, incorrect rounding, or quoting more figures for the answer than they had used in the calculation.