Numeracy Higher November 2024 Unit 2 Q5
5. Morgan is building 2 new houses on a plot of land.
Diagram not drawn to scale
Drainage pipes need to be laid from the 2 new houses to the connection point shown on the diagram.
Morgan cannot measure the lengths of the drainage pipes needed, as he does not have access to the connection point.
He has drawn the following diagram of the drainage pipes, showing some angles and lengths that he has been able to measure.
Diagram not drawn to scale
Use Morgan’s diagram to calculate:
- the length of the drainage pipe needed from New House 1
- the length of the drainage pipe needed from New House 2.
[7]
Length of the drainage pipe needed from New House 1 is …………… m
Length of the drainage pipe needed from New House 2 is …………… m
From the formula list: In any triangle \(ABC\): Sine rule \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\); Cosine rule \(a^2 = b^2 + c^2 - 2bc\cos A\)
| Scheme | Marks | Comments |
|---|---|---|
| Strategy of attempting to use the cosine rule followed by sine rule | S1 | |
| (Length of pipe from new house 1 =) \(\sqrt{8.1^2 + 10.6^2 - 2 \times 8.1 \times 10.6 \times \cos 73}\) (=\(\sqrt{127.763. \ldots}\)) | M2 | M1 for \(8.1^2 + 10.6^2 - 2\times 8.1\times 10.6\times \cos 73\) |
| = 11.3(032…) (m) | A1 | CAO |
| (Length of pipe from new house 2 =) \(\dfrac{11.3(032\ldots)}{\sin 54}\) × sin80 | M2 | Or full alternative method FT ‘their derived 11.3(032…)’ M1 for \(\dfrac{\text{length}}{\sin 80}\) = \(\dfrac{11.3(032\ldots)}{\sin 54}\) or equivalent |
| = 13.75(…) or 13.76 or 13.8 (m) | A1 | Allow 13.7 (m) from correct working Must come from M2 |