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Date May 2022 Marks available 2 Reference code 22M.1.SL.TZ2.3
Level Standard Level Paper Paper 1 Time zone Time zone 2
Command term Justify and Determine Question number 3 Adapted from N/A

Question

The owner of a convenience store installs two security cameras, represented by points C1 and C2. Both cameras point towards the centre of the store’s cash register, represented by the point R.

The following diagram shows this information on a cross-section of the store.

The cameras are positioned at a height of 3.1m, and the horizontal distance between the cameras is 6.4m. The cash register is sitting on a counter so that its centre, R, is 1.0m above the floor.

The distance from Camera 1 to the centre of the cash register is 2.8m.

Determine the angle of depression from Camera 1 to the centre of the cash register. Give your answer in degrees.

[2]
a.

Calculate the distance from Camera 2 to the centre of the cash register.

[4]
b.

Without further calculation, determine which camera has the largest angle of depression to the centre of the cash register. Justify your response.

[2]
c.

Markscheme

sinθ=2.12.8   OR   tanθ=2.11.85202           (M1)

θ=48.6°   48.5903°           A1

 

[2 marks]

a.

METHOD 1

2.82-2.12   OR   2.8cos48.5903   OR   2.1tan48.5903           (M1)


Note: Award M1 for attempt to use Pythagorean Theorem with 2.1 seen or for attempt to use cosine or tangent ratio.


1.85m   1.85202           (A1)


Note: Award the M1A1 if 1.85 is seen in part (a).


6.4-1.85202

4.55m   4.54797           (A1)


Note: Award A1 for 4.55 or equivalent seen, either as a separate calculation or in Pythagorean Theorem.


4.547972+2.12

5.01m   5.00939m           A1

 

METHOD 2

attempt to use cosine rule           (M1)

c2= 2.82+6.42-22.86.4cos48.5903           (A1)(A1)


Note: Award A1 for 48.5903...° substituted into cosine rule formula, A1 for correct substitution.


c=  5.01m   5.00939m           A1

 

[4 marks]

b.

camera 1 is closer to the cash register (than camera 2 and both cameras are at the same height on the wall)           R1

the larger angle of depression is from camera 1           A1


Note: Do not award R0A1. Award R0A0 if additional calculations are completed and used in their justification, as per the question. Accept “1.85<4.55” or “2.8<5.01” as evidence for the R1.

 

[2 marks]

c.

Examiners report

Many candidates calculated the angle from vertical rather than the angle of depression.

a.

Candidates could successfully use their vertical angle from (a) or other correct trigonometry, such as Pythagorean theorem or cosine rule, to find the distance from camera 2 to the cash register. This question is a good example of how premature rounding can affect a final answer, and some had an inaccurate final answer because they had rounded intermediate values.

b.

Many provided reasonable justification for their response, even though they often followed correct reasoning with an incorrect conclusion about the larger angle of depression.

c.

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.2—2d and 3d trig
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Topic 3—Geometry and trigonometry » SL 3.3—Angles of elevation and depression
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