Numeracy Higher June 2024 Unit 2 Q6
6.
C-clips are used in motors and turbines as a type of fastener.
Clip Zone makes metal C-clips.
They have a uniform cross-section, as shown below.
Diagrams not drawn to scale
\(O\) is the centre of the circular arcs \(AC\) and \(BD\).
\(OAB\) and \(OCD\) are straight lines.
\(OA = OC =\) 50 mm.
\(AB = CD =\) 4 mm.
The C-clips have a uniform thickness of 2 mm.
C-clips are made by melting down metal bars and re-casting them.
Calculate the number of C-clips that can be made from a metal bar with a volume of 1 500 000 mm\(^3\).
You must show all your working. [5]
Clip Zone makes four types of fastener.
The table below shows the number of each type produced each week.
| Type of fastener | C-clips | Jubilee clips | Screws | Bolts |
|---|---|---|---|---|
| Number made each week | 23 000 | 11 000 | 70 000 | 45 000 |
Clip Zone takes a sample of these 149 000 fasteners to check the quality of its products.
A stratified sample of 1500 fasteners is taken, based on the type of fastener.
Calculate the number of each type of fastener that should be included in the sample.
You must show all your working. [3]
| Type of fastener | C-clips | Jubilee clips | Screws | Bolts |
|---|---|---|---|---|
| Number in the sample | …………… | …………… | …………… | …………… |
| Scheme | Mark | Comments |
|---|---|---|
| Accept use of values of \(\boldsymbol{\pi}\) from 3.14 to 3.142 Note: Accept (54\(^2\) – 50\(^2\)) written as (54 + 50) × (54 – 50) | ||
| \(\dfrac{360 - 38}{360}\) × \(\boldsymbol{\pi}\) × (54\(^2\) – 50\(^2\)) (× 2) or equivalent (= 1168.35 to 1169.11 or 16744\(\boldsymbol{\pi}\)/45) | M2 | M1 for one of the following:
|
| = 2336.7 to 2338.21 or 33488\(\boldsymbol{\pi}\)/45 (mm\(^3\)) | A1 | CAO If no marks yet awarded, SC1 for any one of the following
|
| (Number of C-clips =) \(\dfrac{1\,500\,000}{2336.7 \text{ to } 2338.21}\) | M1 | FT ‘their 2336.7 to 2338.21’ provided at least 1 mark previously awarded |
| = 641 (C-clips) | A1 | Only FT from M2M1 or M1M1 previously awarded On FT, should be an integer obtained from the truncation of their answer |
Alternative
6(a) Alternative method for the first 3 marks:
| Scheme | Mark | Comments |
|---|---|---|
| \(\dfrac{360 - 38}{360}\) × \(\dfrac{(2\times\boldsymbol{\pi}\times 54 + 2\times\boldsymbol{\pi}\times 50)}{2}\) × (54 – 50) (× 2) | M2 | Accept use of values of \(\boldsymbol{\pi}\) from 3.14 to 3.142 M1 for one of the following:
|
| = 2336.7 to 2338.21 (mm\(^3\)) | A1 | CAO If no marks yet awarded, SC1 for any one of the following:
|
| Scheme | Mark | Comments |
|---|---|---|
| 1500 × \(\dfrac{\text{(number of a type of fastener made)}}{149\,000}\) | M1 | Sight of this calculation for any type of fastener Allow e.g. (C-clips) \(\dfrac{15.4(\ldots)}{100}\) × 1500 |
| (List of unrounded answers) 231.5(436…), 110.7(382…), 704.6(979…), 453(.0201…) | A1 | OR A1 for 232, 111, 705, 453 Implies the award of M1 |
| (Number in sample =) 231, 111, 705, 453 | A1 | Award M1A1A1 for 231, 111, 705, 453 provided no incorrect work seen If M1A0 awarded, A1 on FT from their unrounded answers for equivalent difficulty i.e. provided:
|