Numeracy Higher November 2020 Unit 1 Q7
7. A box for mints is to be made in the shape of a hexagonal prism.
The cross-section of the box is a regular hexagon.
The volume of the box must be greater than 230 000 mm3.
Diagrams not drawn to scale
From the formula list: Volume of prism = area of cross-section × length
Using the measurements above, show that this would make a suitable box for the mints.
You must show all your working. [5]
In all alternative methods for answering this question accept alternative working in cm, if place value error in conversion of units penalise -1 once only
| Scheme | Marks | Comments |
|---|---|---|
| (Area of cross-section) 6 × ½ × 30 × (52 ÷ 2) | M3 | M2 for ½ × 30 × (52 ÷ 2) (= 390) M1 for any use of 52 ÷ 2 (= 26) (May be embedded) |
| 2340 (mm2) | A1 | |
| (Volume of the box) 234000 (mm3) OR for a comparison 2340 (mm2) > 2300 (mm2) | A1 | FT ‘their 2340’ × 100 correctly evaluated provided at least M2 previously awarded |
Alternative method (trapezia)
| Scheme | Marks | Comments |
|---|---|---|
| (Area of cross-section) 2 × ½ × (52÷2) ×(30 + 2×30) | M3 | M2 for ½ × (52÷2) ×(30 + 2×30) (= 1170) M1 for use of 52 ÷ 2 (= 26) |
| 2340 (mm2) | A1 | |
| (Volume of the box) 234000 (mm3) OR for a comparison 2340 (mm2) > 2300 (mm2) | A1 | FT ‘their 2340’ × 100 correctly evaluated provided at least M2 previously awarded |
Alternative method (½absinC)
| Scheme | Marks | Comments |
|---|---|---|
| (Area of cross-section) 6 × ½ × 30 × 30 × \(\dfrac{\sqrt{3}}{2}\) | M3 | M2 for ½ × 30 × 30 × \(\dfrac{\sqrt{3}}{2}\) M1 for (6 ×) ½ × 30 × 30 ×sin 60° |
| 1350√3 or 2338(.2... mm2) or 2340 (mm2) | A1 | |
| (Volume of box) 233820 mm3 or 234000 (mm3) OR for a comparison 2338(.2 mm2) > 2300 (mm2) | A1 | FT ‘their 2340’ × 100 correctly evaluated provided at least M2 previously awarded |
Alternative method (triangle area)
| Scheme | Marks | Comments |
|---|---|---|
| (Area of triangle) ½ × 30 × (52 ÷ 2) | M2 | (= 390) |
| (Minimum area of triangle required) 2300 ÷ 6 | M1 | |
| 383(.33...) | A1 | |
| Comparison 390 > 383(.33...) | A1 |