Numeracy Higher November 2021 Unit 1 Q9
9. A bridge in Pontrhydyfen, located in the Afan Valley, was built in the 19th century.
From 2016 to 2017, restoration work was carried out on most of the surfaces of the bridge.
The diagram below shows a simplified version of the bridge.
The semi-circular arches each have a diameter of 20 m.
The entire length of the bridge has a uniform width of 4 m.
All the pillars are identical cuboids.
Diagram not drawn to scale
All of the surfaces were restored except the shaded areas at the top and at both ends of the bridge.
(Note, in the diagram, only one of the ends can be seen.)
Calculate the total surface area that was restored.
Give your answer, in terms of \(\pi\), in its simplest form. [7]
Total surface area that was restored = …………… m2
| Scheme | Marks | Comments |
|---|---|---|
| 9. (Total SA of pillars =) 14 × 4 × 18 (=1008) | B1 | |
| (Total SA of other flat faces =) 100 × 12 × 2 − \(\dfrac{\pi \times 10^{2}}{2}\) × 8 or equivalent (=2400 − 400π) | B2 | B1 for 100×12 − \(\dfrac{\pi \times 10^{2}}{2}\) × 4 (=1200 − 200π) OR B1 for \(\dfrac{\pi \times 10^{2}}{2}\) × 8 (=400π) |
| (Total curved SA =) \(\dfrac{\pi \times 20}{2}\) × 4 × 4 or equivalent (=160π) | B1 | |
| (Total surface area =) 14×4×18 + 100×12×2 + \(\dfrac{\pi \times 20}{2}\)×4×4 − \(\dfrac{\pi \times 10^{2}}{2}\) × 8 or equivalent | M2 | M1 for the addition/subtraction of at least 3 correct terms |
| = 3408 − 240π (m2) | A1 | CAO. ISW |
9. Alternative method:
| Scheme | Marks | Comments |
|---|---|---|
| (Area of front and back faces =) 100 × 26 × 2 − 4 × (\(\dfrac{\pi \times 10^{2}}{2}\) + 14×20) × 2 or equivalent (= 5200 − 400π – 2240) | B2 | B1 for 100×26 − 4×(\(\dfrac{\pi \times 10^{2}}{2}\) + 14×20) OR B1 for 4 × (\(\dfrac{\pi \times 10^{2}}{2}\) + 14×20) × 2 |
| (Total curved SA =) \(\dfrac{\pi \times 20}{2}\) × 4 × 4 or equivalent (=160π) | B1 | |
| (Inside of pillars =) 14 × 4 × 8 (=448) | B1 | |
| (Total surface area =) 100×26×2 − 4×(\(\dfrac{\pi \times 10^{2}}{2}\) + 14×20)×2 + \(\dfrac{\pi \times 20}{2}\) ×4×4 + 14 × 4 × 8 or equivalent | M2 | M1 for the addition/subtraction of at least 4 correct terms |
| = 3408 − 240π (m2) | A1 | CAO. ISW |