Date | November 2018 | Marks available | 2 | Reference code | 18N.2.SL.TZ0.S_7 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Show that | Question number | S_7 | Adapted from | N/A |
Question
A communication tower, T, produces a signal that can reach cellular phones within a radius of 32 km. A straight road passes through the area covered by the tower’s signal.
The following diagram shows a line representing the road and a circle representing the area covered by the tower’s signal. Point R is on the circumference of the circle and points S and R are on the road. Point S is 38 km from the tower and RŜT = 43˚.
Let SR = x. Use the cosine rule to show that x2−(76cos43∘)x+420=0.
Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
recognizing TR =32 (seen anywhere, including diagram) A1
correct working A1
eg 322=x2+382−2(x)(38)cos43∘, 1024=1444+x2−76(x)cos43∘
x2−(76cos43∘)x+420=0 AG N0
[2 marks]
Note: There are many approaches to this question, depending on which triangle the candidate has used, and whether they used the cosine rule and/or the sine rule. Please check working carefully and award marks in line with the markscheme.
METHOD 1
correct values for x (seen anywhere) A1A1
x = 9.02007, 46.5628
recognizing the need to find difference in values of x (M1)
eg 46.5 − 9.02, x1−x2
37.5427
37.5 (km) A1 N2
METHOD 2
correct use of sine rule in ΔSRT
eg sinS∧RT38=sin43∘32, S∧RT = 54.0835° (A1)
recognizing isosceles triangle (seen anywhere) (M1)
eg ˆT=180∘−2⋅54.0835∘, two sides of 32
correct working to find distance A1
eg √322+322−2⋅32⋅32cos(180∘−2⋅54.0835∘),
sin71.8329∘d=sin54.0835∘32, 322=322+x2−2⋅32xcos(0.944)
37.5427
37.5 (km) A1 N2
[4 marks]