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Date	November 2021	Marks available	2	Reference code	21N.2.SL.TZ0.4
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Determine	Question number	4	Adapted from	N/A

Question

A large water reservoir is built in the form of part of an upside-down right pyramid with a horizontal square base of length $80$ metres. The point $C$ is the centre of the square base and point $V$ is the vertex of the pyramid.

The bottom of the reservoir is a square of length $60$ metres that is parallel to the base of the pyramid, such that the depth of the reservoir is $6$ metres as shown in the diagram.

The second diagram shows a vertical cross section, $MNOPC$ , of the reservoir.

Every day $80 m^{3}$ of water from the reservoir is used for irrigation.

Joshua states that, if no other water enters or leaves the reservoir, then when it is full there is enough irrigation water for at least one year.

Find the angle of depression from $M$ to $N$ .

[2]

Find $CV$ .

[2]

b.i.

Hence or otherwise, show that the volume of the reservoir is $29 600 m^{3}$ .

[3]

b.ii.

By finding an appropriate value, determine whether Joshua is correct.

[2]

To avoid water leaking into the ground, the five interior sides of the reservoir have been painted with a watertight material.

Find the area that was painted.

[5]

Markscheme

$\tan (θ) = \frac{6}{10}$ (M1)

$(θ =) 31.0 ° (30.9637 \dots °)$ OR $0.540 (0.540419 \dots)$ A1

[2 marks]

$(CV =) 40 \tan (θ)$ OR $(CV =) 4 \times 6$ (M1)

Note: Award (M1) for an attempt at trigonometry or similar triangles (e.g. ratios).

$(CV =) 24 m$ A1

[2 marks]

b.i.

$(V =) \frac{1}{3} 80^{2} \times 24 - \frac{1}{3} 60^{2} \times 18$ M1A1A1

Note: Award M1 for finding the difference between the volumes of two pyramids, A1 for each correct volume expression. The final A1 is contingent on correct working leading to the given answer.
If the correct final answer is not seen, award at most M1A1A0. Award M0A0A0 for any height derived from $V = 29 600$ , including $18.875$ or $13.875$ .

$(V =) 29 600 m^{3}$ AG

[3 marks]

b.ii.

METHOD 1

$(\frac{29 600}{80} =) 370$ (days) A1

$(370 > 366)$ Joshua is correct A1

Note: Award A0A0 for unsupported answer of “Joshua is correct”. Accept $1.01 \dots > 1$ for the first A1 mark.

METHOD 2

$80 \times 366 = 29280 m^{3}$ OR $80 \times 365 = 29200 m^{3}$ A1

$(29280 < 29600)$ Joshua is correct A1

Note: The second A1 can be awarded for an answer consistent with their result.

[2 marks]

height of trapezium is $\sqrt{10^{2} + 6^{2}} (= 11.6619 \dots)$ (M1)

area of trapezium is $\frac{80 + 60}{2} \times \sqrt{10^{2} + 6^{2}} (= 816.333 \dots)$ (M1)(A1)

$(S A =) 4 \times (\frac{80 + 60}{2} \times \sqrt{10^{2} + 6^{2}}) + 60^{2}$ (M1)

Note: Award M1 for adding $4$ times their ( $MNOP$ ) trapezium area to the area of the ( $60 \times 60$ ) base.

$(S A =) 6870 m^{2} (6865.33 m^{2})$ A1

Note: No marks are awarded if the correct shape is not identified.

[5 marks]

Examiners report

Finding the angle of depression from $M$ to $N$ proved problematic for many. Some chose the angle inclined to the vertical or attempted a less efficient method such as solving triangle $NMP$ . Most candidates attempted to find $CV$ through trigonometry rather than similar triangles. Consistent with past examination sessions, it is clear that candidate do not always understand the demands of a 'show that' question. Working backwards to verify the given volume of $29600 m^{3}$ accrued no marks. Various approaches were used, with most candidates earning at least one mark for one correct volume seen. Part (c) was accessible to all. It was surprising to see some candidates identify $360$ as the number of days in a calendar year. Part (d) was a true high-order discriminator, proving to be challenging for even very capable candidates. Some candidates found the surface area of a shape/object that is not part of the reservoir, while the majority incorrectly identified the height of trapezoid $MPON$ as $6$ .

b.i.

b.ii.

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.1—3d space, volume, angles, midpoints

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Hide related questions

Topic 3—Geometry and trigonometry » SL 3.2—2d and 3d trig

Topic 3—Geometry and trigonometry » SL 3.3—Angles of elevation and depression

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

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