Date | May 2014 | Marks available | 1 | Reference code | 14M.2.sl.TZ2.2 |
Level | SL only | Paper | 2 | Time zone | TZ2 |
Command term | Calculate | Question number | 2 | Adapted from | N/A |
Question
A cross-country running course consists of a beach section and a forest section. Competitors run from A to B, then from B to C and from C back to A.
The running course from A to B is along the beach, while the course from B, through C and back to A, is through the forest.
The course is shown on the following diagram.
Angle ABC is 110∘.
It takes Sarah 5 minutes and 20 seconds to run from A to B at a speed of 3.8 ms−1.
Using ‘distance = speed × time’, show that the distance from A to B is 1220 metres correct to 3 significant figures.
The distance from B to C is 850 metres. Running this part of the course takes Sarah 5 minutes and 3 seconds.
Calculate the speed, in ms−1, that Sarah runs from B to C.
The distance from B to C is 850 metres. Running this part of the course takes Sarah 5 minutes and 3 seconds.
Calculate the distance, in metres, from C to A.
The distance from B to C is 850 metres. Running this part of the course takes Sarah 5 minutes and 3 seconds.
Calculate the total distance, in metres, of the cross-country running course.
The distance from B to C is 850 metres. Running this part of the course takes Sarah 5 minutes and 3 seconds.
Find the size of angle BCA.
The distance from B to C is 850 metres. Running this part of the course takes Sarah 5 minutes and 3 seconds.
Calculate the area of the cross-country course bounded by the lines AB, BC and CA.
Markscheme
3.8×320 (A1)
Note: Award (A1) for 320 or equivalent seen.
=1216 (A1)
=1220 (m) (AG)
Note: Both unrounded and rounded answer must be seen for the final (A1) to be awarded.
[2 marks]
850303 (ms−1) (2.81, 2.80528…) (A1)(G1)
[1 mark]
AC2=12202+8502−2(1220)(850)cos110∘ (M1)(A1)
Note: Award (M1) for substitution into cosine rule formula, (A1) for correct substitutions.
AC=1710 (m) (1708.87…) (A1)(G2)
Notes: Accept 1705 (1705.33…).
[3 marks]
1220+850+1708.87… (M1)
=3780 (m) (3778.87…) (A1)(ft)(G1)
Notes: Award (M1) for adding the three sides. Follow through from their answer to part (c). Accept 3771 (3771.33…).
[2 marks]
sinC1220=sin110∘1708.87… (M1)(A1)(ft)
Notes: Award (M1) for substitution into sine rule formula, (A1)(ft) for correct substitutions. Follow through from their part (c).
C=42.1∘ (42.1339…) (A1)(ft)(G2)
Notes: Accept 41.9∘,42.0∘,42.2∘,42.3∘.
OR
cosC=1708.87…2+8502−122022×1708.87…×850 (M1)(A1)(ft)
Notes: Award (M1) for substitution into cosine rule formula, (A1)(ft) for correct substitutions. Follow through from their part (c).
C=42.1∘ (42.1339…) (A1)(ft)(G2)
Notes: Accept 41.2∘,41.8∘,42.4∘.
[3 marks]
12×1220×850×sin110∘ (M1)(A1)(ft)
OR
12×1708.87…×850×sin42.1339…∘ (M1)(A1)(ft)
OR
12×1220×1708.87…×sin27.8661…∘ (M1)(A1)(ft)
Note: Award (M1) for substitution into area formula, (A1)(ft) for correct substitution.
=487000 m2 (487230… m2) (A1)(ft)(G2)
Notes: The answer is 487000 m2, units are required.
Accept 486000 m2 (485633… m2).
If workings are not shown and units omitted, award (G1) for 487000 or 486000.
Follow through from parts (c) and (e).
[3 marks]