Numeracy Higher November 2018 Unit 1 Q12
12. An architect has been asked to design a square-based glass pyramid that is to be built in a park.
The vertex at the top of the pyramid will be directly above the centre of the square base.
The edges of the pyramid will be made from steel.
Each sloping face will be made from glass.
Diagram not drawn to scale
The architect first plans to make a scale model of the pyramid.
The scale model will have base sides of length 14 cm and a vertical height of 8 cm.
The architect has drawn the following diagram of the model.
Diagram not drawn to scale
Calculate the length of each sloping edge of the model.
Give your answer in the form \(a\sqrt{b}\), where a is an integer and b is a prime number. [6]
The area of each glass face on the scale model is 74 cm2.
The full-size pyramid is to have a vertical height of 1·6 metres.
Use these values to calculate the area of each glass face on the full-size pyramid.
Give your answer in cm2. [5]
| Scheme | Marks | Comments |
|---|---|---|
| (Half base diagonal2 =) 72 + 72 OR Half base diagonal2 = 98 OR (Half base diagonal =) \(\sqrt{98}\) OR Half base diagonal = ½ \(\sqrt{14^2 + 14^2}\) | M1 | Allow M1 for an appropriate 72 + 82 OR 113 OR \(\sqrt{113}\) |
| (Slant edge =) \(\sqrt{98 + 8^2}\) OR \(\sqrt{7^2 + 7^2 + 8^2}\) | M2 | M1 for 98 + 82 OR 72 + 72 + 82 |
| Slant edge = \(\sqrt{162}\) | A1 | CAO |
| = \(9\sqrt{2}\) | B2 | For B2, FT ‘their derived 162’ provided their 'b' is a prime number For B1, FT 'their derived 162' B1 for writing 162 as a product of 2 or more factors where one of the factors OR the product of a pair of their factors is a square number e.g. 2 × 81, 3 × 3 × 18, OR B1 for writing \(\sqrt{162}\) as a product of 2 or more factors where one of the factors OR the product of a pair of their factors is a whole number e.g. \(\sqrt{2} \times \sqrt{9} \times \sqrt{9}\) |
| Scheme | Marks | Comments |
|---|---|---|
| (Area factor =) \(\left(\dfrac{160}{8}\right)^2\) or equivalent | M2 | M1 for \(\dfrac{160}{8}\) or equivalent OR sight of 20 OR M1 for \(\left(\dfrac{160}{8}\right)^2\) with place value errors e.g. 22 or 0·04 |
| = 400 | A1 | CAO. May be implied in further working |
| (Area of glass =) 400 × 74 | M1 | FT ‘their \(\left(\dfrac{160}{8}\right)^2\)’ or ‘their \(\left(\dfrac{1{\cdot}6}{8}\right)^2\)’ Must be from an area scale factor |
| = 29 600 (cm2) | A1 | CAO |