Numeracy Higher November 2023 Unit 1 Q9
9. Akago is a food delivery company.
Akago wants to:
- increase the number of deliveries to 240 houses
- shorten the total delivery time to 3 hours.
Calculate the number of delivery vans that would be needed to deliver food to 240 houses in 3 hours.
You can assume that all vans deliver food to all the houses at the same rate. [3]
Number of delivery vans needed = ……………
It is made from rectangles, connected by sectors of circles.
The company prints its logo on its delivery boxes.
The logo is shown below.
Diagram not drawn to scale
Calculate the area of the logo that is printed on the company’s delivery boxes.
Give your answer in terms of \(\pi\) in its simplest form. [6]
Area of the logo = …………… cm2
Diagram not drawn to scale
The larger square has an area of 45 cm2.
The smaller square has an area of 5 cm2.
Calculate the total height of the two stacked food boxes, as shown in the diagram.
Give your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers, and \(b\) is as small as possible. [3]
| Scheme | Marks | Comments | ||||||
|---|---|---|---|---|---|---|---|---|
| 5 × \(\dfrac{240}{100}\) or × 2.4 or equivalent | M1 | A table method altering all 3 in the same manner at the same time is M0 M marks may be seen in either order e.g.
| ||||||
| × \(\dfrac{4}{3}\) or × 1.333… or equivalent | M1 | FT from M0 previously awarded Must be from use of 5 e.g. if this calculation is performed first
| ||||||
| = 16 (delivery vans) | A1 | CAO |
9(a) Alternative method 1:
| Scheme | Marks | Comments |
|---|---|---|
| \(\dfrac{100}{4\times5}\) (=5 houses per hour per van) | M1 | |
| \(\dfrac{240}{100\div(4\times5)\times3}\) or \(\dfrac{240}{5\times3}\) | m1 | |
| = 16 (delivery vans) | A1 | CAO |
9(a) Alternative method 2:
| Scheme | Marks | Comments | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 | ||||||||||
| 5 × \(\dfrac{240}{75}\) OR 5 × \(\dfrac{9.6}{3}\) or 5 × 3.2 or equivalent | M1 | FT from 3 (time), n houses, 5 vans for 5 × \(\dfrac{240}{n}\) FT from n (time), 240 houses, 5 vans for 5 × \(\dfrac{n}{3}\) | |||||||||
| = 16 (delivery vans) | A1 | CAO |
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\dfrac{\text{angle}}{360}\times\pi\times3^{2}\) or equivalent | B1 | Allow use of \(\pi\) = 3.14 to 3.142 for B and M marks, but not for A marks For any of the angles 140(°), 110(°), 70(°), 140+110+70 or 320 |
| \(\dfrac{140}{360}\times\pi\times3^{2} + \dfrac{110}{360}\times\pi\times3^{2} + \dfrac{70}{360}\times\pi\times3^{2}\) or equivalent (\(7\pi/2\)) (\(11\pi/4\)) (\(7\pi/4\)) | M2 | e.g. \(\dfrac{(140+110+70)}{360}\times\pi\times3^{2}\) or \(\dfrac{320}{360}\times\pi\times3^{2}\) or \(8\pi\) M1 for the sum of any 2 correct terms OR M1 for \(\dfrac{x}{360}\times\pi\times3^{2}\) where \(300 \leqslant x \lt 360\) if 140+110+70 not seen The award of M2 or M1 implies the previous B1 |
| + 3×12×3 + 22×3 | m1 | FT from M1 |
| = \(8\pi + 174\) (cm2) | A2 | CAO. Mark final answer A1 (does not depend on m1 being awarded) for any one of the following seen
|
| Scheme | Marks | Comments |
|---|---|---|
| \(\sqrt{9}\times\sqrt{5} + \sqrt{5}\) or \(3\sqrt{5} + \sqrt{5}\) | M2 | M1 for \(\sqrt{45} + \sqrt{5}\) or M1 for sight of \(3\sqrt{5}\) |
| = \(4\sqrt{5}\) (cm) | A1 | CAO |