Write down the length of VA in metres.

Sketch the triangle VCA showing clearly the length of VC and the size of angle VCA.

Show that the height of the pyramid is 18.0 metres correct to 3 significant figures.

Calculate the length of AC in metres.

Show that the length of BC is 19.1 metres correct to 3 significant figures.

Calculate the volume of the tower.

To calculate the cost of air conditioning, engineers must estimate the weight of air in the tower. They estimate that 90 % of the volume of the tower is occupied by air and they know that 1 m³ of air weighs 1.2 kg.

Calculate the weight of air in the tower.

22.5 (m)     (A1)

[1 mark]

     (A1)

[1 mark]

h = 22.5     sin 53.1°     (M1)
= 17.99     (A1)
= 18.0     (AG)

Note: Unrounded answer must be seen for (A1) to be awarded.

Accept 18 as (AG).

[2 marks]

\({\text{AC}} = 2\sqrt {{{22.5}^2} - {{17.99...}^2}} \)     (M1)(M1)

Note: Award (M1) for multiplying by 2, (M1) for correct substitution into formula.

OR

AC = 2(22.5)cos53.1°     (M1)(M1)

Notes: Award (M1) for correct use of cosine trig ratio, (M1) for multiplying by 2.

OR

AC² = 22.5² + 22.5² – 2(22.5)(22.5) cos73.8°     (M1)(A1)

Note: Award (M1) for substituted cosine formula, (A1) for correct substitutions.

OR

\(\frac{{{\text{AC}}}}{{\sin (73.8^\circ )}} = \frac{{22.5}}{{\sin (53.1^\circ )}}\)     (M1)(A1)

Note: Award (M1) for substituted sine formula, (A1) for correct substitutions.

AC = 27.0     (A1)(G2)

[3 marks]

\({\text{BC}} = \sqrt {{{13.5}^2} + {{13.5}^2}} \)     (M1)

= 19.09     (A1)

= 19.1     (AG)

OR

x² + x² = 27²     (M1)

2x² = 27²     (A1)

BC = 19.09…     (A1)

= 19.1     (AG)

Notes: Unrounded answer must be seen for (A1) to be awarded.

[2 marks]

Volume = Pyramid + Cuboid

\( = \frac{1}{3}(18)({19.1^2}) + (108)({19.1^2})\)     (A1)(M1)(M1)

Note: Award (A1) for 108, the height of the cuboid seen. Award (M1) for correctly substituted volume of cuboid and (M1) for correctly substituted volume of pyramid.

= \(41\,588\)     (41\(\,\)553 if 2(13.5²) is used)

= \(41\,600\) m³     (A1)(ft)(G3)

[4 marks]

Weight of air = \(41\,600 \times 1.2 \times 0.9\)     (M1)(M1)

= \(44\,900{\text{ kg}}\)     (A1)(ft)(G2)

Note: Award (M1) for their part (e) × 1.2, (M1) for × 0.9.

Award at most (M1)(M1)(A0) if the volume of the cuboid is used.

[3 marks]

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.

This question also caused many problems for the candidature. There seems to be a lack of ability in visualising a problem in three dimensions – clearly, further exposure to such problems is needed by the students. Further, as in question 2, the final two parts of the question were independent of those preceding them; many candidates did not reach these parts, though for some, these were the only parts of the question attempted. There is also a lack of awareness of the appropriate volume formula on the formula sheet to use.