Numeracy Higher June 2017 Unit 2 Q14
14. The diagram shows a 5 m wide section of road that has a uniform gradient.
The shaded area represents level ground.
Two cyclists, Delyth and Ioan, approach this section of road.
Diagram not drawn to scale
Delyth cycles straight up the middle of the road as shown by the arrow.
Ioan thinks this section of road is too steep to cycle straight up, so he decides to cycle from A to B in a straight line.
(a) How far does Ioan cycle in going from A to B? [6]
(b) Show that Ioan’s route up this section of road is less steep than Delyth’s route.
You must show all your working. [3]
You must show all your working. [3]
| Scheme | Marks | Comments |
|---|---|---|
| (Hyp of triangle =) 1 ÷ (sin7∙1(°)) | M2 | Or equivalent M1 for sin7∙1(°) = 1 ÷ (Hyp of triangle) |
| = 8∙0(9051…) or 8∙1 (m) | A1 | Accept 8 (m) from correct working |
| (AB2 =) 52 + 8∙0(9051…)2 | M1 | FT their 8∙0(9051…) provided trigonometry attempted |
| AB2 = 90∙4(564…) or (AB =) √90∙4(564…) | A1 | FT their rounded 8∙0(9051…) |
| (AB =) 9∙5(108…) (m) | A1 | Do not accept 9∙4 (m) from use of 8 (m) Needs to be correct to 1 d.p. FT from previous M1 for the correctly evaluated square root of ‘their 90∙4(564…) ’ provided ‘their answer’ > ‘their 8∙09(051…)’ Alternative method: Base of triangle = 1/tan7∙1 (=8∙0284…) M1 52 + 8∙0(284…)2 (= 89∙456…) M1 Base diagonal = 9∙4(581…) or 9∙5 (m) A1 12 + 9∙4(581…)2 M1 FT ‘their rounded 9∙4(581…)’ AB2 = 90∙4(564…) or (AB =) √90∙4(564…) A1 (AB =) 9∙5(108…) (m) A1 Needs to be correct to 1 d.p. FT from previous M1 for the correctly evaluated square root of ‘their 90∙4(564…)’ provided ‘their answer’ > ‘their 8∙09(051…)’ |
| Scheme | Marks | Comments |
|---|---|---|
| sin−1 (1/9∙5(108…)) | M2 | FT ‘their 9∙5(108…)’ M1 for sin(angle) = 1/9∙5(108…) |
| = 6(∙0354…)(°) | A1 | Needs to be an answer that is < 7∙1 Needs to be correct to 1 d.p. Do not penalise premature rounding on FT if already penalised in (a) Alternative method: tan−1 (1/9∙4(581…)) M2 = 6∙0(354…)(°) A1, OR cos−1 (9∙4(581…) / 9∙5(108…)) M2 = 6∙0(354…)(°) A1 OR Alternative method: B1 for ‘Delyth’s route is going up 1(m) in (travelling) 8∙0(9051… m).’ OR ‘The gradient of Delyth’s route is (1/8∙0(284…)) 0∙12(455..)’ B1 for ‘Ioan’s route is going up 1(m) in (travelling) 9∙5(108…m)’ OR ‘The gradient of Ioan’s route is (1/9∙4(581…)) 0∙10(572…)’ B1 for ‘So Ioan’s route is less steep.’ This B1 is dependent on previous B1B1 being awarded and provided Ioan’s gradient is less steep than Delyth’s FT their values from (a) |