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Date	May 2021	Marks available	2	Reference code	21M.2.SL.TZ1.8
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	8	Adapted from	N/A

Question

Two straight fences meet at point $A$ and a field lies between them.

A horse is tied to a post, $P$ , by a rope of length $r$ metres. Point $D$ is on one fence and point $E$ is on the other, such that $PD = PE = PA = r$ and $D \hat{P} E = θ$ radians. This is shown in the following diagram.

The length of the arc $DE$ shown in the diagram is $28 m$ .

A new fence is to be constructed between points $B$ and $C$ which will enclose the field, as shown in the following diagram.

Point $C$ is due west of $B$ and $AC = 800 m$ . The bearing of $B$ from $A$ is $195 °$ .

Write down an expression for $r$ in terms of $θ$ .

[1]

a.

Show that the area of the field that the horse can reach is $\frac{392}{θ^{2}} (θ + \sin θ)$ .

[4]

b.

The area of field that the horse can reach is $460 m^{2}$ . Find the value of $θ$ .

[2]

c.

Hence, find the size of $D \hat{A} E$ .

[2]

d.

Find the size of $A \hat{B} C$ .

[2]

e.i.

Find the length of new fence required.

[3]

e.ii.

Markscheme

$r = \frac{28}{θ}$ A1

[1 mark]

a.

recognising sum of area of sector and area of triangle required (M1)

$\frac{1}{2} r^{2} θ + \frac{1}{2} r \times r \times \sin (π - θ) (= \frac{r^{2}}{2} (θ + \sin (π - θ))$ A1

$\sin (π - θ) = \sin θ$ (substitution seen anywhere) A1

$\frac{1}{2} {(\frac{28}{θ})}^{2} θ + \frac{1}{2} {(\frac{28}{θ})}^{2} \sin θ$ OR $\frac{1}{2} {(\frac{28}{θ})}^{2} (θ + \sin θ)$ A1

area $= \frac{392}{θ^{2}} (θ + \sin θ)$ AG

[4 marks]

b.

$\frac{392}{θ^{2}} (θ + \sin θ) = 460$ (M1)

$θ = 1.43917 \dots$

$θ = 1.44$ A1

[2 marks]

c.

$\frac{π - (π - θ)}{2}$ OR $\frac{θ}{2}$ (M1)

$D \hat{A} E = 0.719588 \dots$

$D \hat{A} E = 0.720$ A1

[2 marks]

d.

$A \hat{B} C = 195 - 180 + 90$ (A1)

$= 105 °$ A1

[2 marks]

e.i.

choosing sine rule (M1)

$\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 105}$ OR $\frac{BC}{sin D \hat{A} E} = \frac{800}{\sin 1.83}$ A1

$BC = 546 (m)$ A1

[2 marks]

e.ii.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

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

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16N.2.SL.TZ0.T_5b:
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16N.2.SL.TZ0.T_5a:
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17N.2.SL.TZ0.T_3b:
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17N.2.SL.TZ0.T_3d.ii:
Find the percentage error in Abdallah’s estimate.
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19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
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State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
19M.2.SL.TZ2.T_2a:
Find the distance from A to C.
19M.2.SL.TZ2.T_2b.i:
Find the size of angle ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}$ .
19M.2.SL.TZ2.T_2b.ii:
Find the size of angle ${\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}}$ .
19M.2.SL.TZ2.T_2c:
Find the size of angle ${\text{A}}\mathop {\text{C}}\limits^ \wedge {\text{D}}$ .
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Calculate the area of triangle EAD.
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17M.2.SL.TZ1.T_2b:
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Calculate the length of CF.
18N.2.SL.TZ0.T_5d:
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Calculate the angle of elevation of T from, M, the midpoint of the side AC.
17M.2.SL.TZ2.T_4a:
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Find the area of the park.
16N.2.SL.TZ0.T_5c:
the area of triangle ABD;
16N.2.SL.TZ0.T_5e:
the length of AP;
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Calculate the length of $BD$ .
20N.2.SL.TZ0.T_3c:
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