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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 ${x^2} - \left( {76\,{\text{cos}}\,43^\circ } \right)x + 420 = 0$ .

[2]

a.

Hence or otherwise, find the total distance along the road where the signal from the tower can reach cellular phones.

[4]

b.

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 ${32^2} = {x^2} + {38^2} - 2\left( x \right)\left( {38} \right){\text{cos}}\,43^\circ$ , $1024 = 1444 + {x^2} - 76\,\left( x \right){\text{cos}}\,43^\circ$

${x^2} - \left( {76\,{\text{cos}}\,43^\circ } \right)x + 420 = 0$ AG N0

[2 marks]

a.

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, ${x_1} - {x_2}$

37.5427

37.5 (km) A1 N2

METHOD 2

correct use of sine rule in ΔSRT

eg $\frac{{{\text{sin}}\,{\text{S}}\mathop {\text{R}}\limits^ \wedge {\text{T}}}}{{38}} = \frac{{{\text{sin}}\,43^\circ }}{{32}}$ , ${{\text{S}}\mathop {\text{R}}\limits^ \wedge {\text{T}}}$ = 54.0835° (A1)

recognizing isosceles triangle (seen anywhere) (M1)

eg $\hat T = 180^\circ - 2 \cdot 54.0835^\circ$ , two sides of 32

correct working to find distance A1

eg $\sqrt {{{32}^2} + {{32}^2} - 2 \cdot 32 \cdot 32\,{\text{cos}}\,\left( {180^\circ - \,2 \cdot 54.0835^\circ } \right)}$ ,

$\frac{{{\text{sin}}\,71.8329^\circ }}{d} = \frac{{{\text{sin}}\,54.0835^\circ }}{{32}}$ , ${32^2} = {32^2} + {x^2} - 2 \cdot 32x\,{\text{cos}}\left( {{\text{0}}{\text{.944}}} \right)$

37.5427

37.5 (km) A1 N2

[4 marks]

b.

Examiners report

[N/A]

a.

[N/A]

b.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.2—2d and 3d trig, sine rule, cosine rule, area

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