Higher June 2023 Paper 1 Q23
23 A frustum is made by removing a small square-based pyramid from a similar large squared-based pyramid as shown in the diagram.
Diagram NOT accurately drawn
The height of the small pyramid is 15 cm.
The height of the large pyramid is 45 cm.
The square base of the large pyramid has side length 39 cm.
Work out the total surface area of the frustum.
Give your answer correct to the nearest whole number.
(5)
| Scheme | Marks |
|---|---|
| (39 ÷ 3)² + 39² or 1521 + 169 (= 1690) | M1 |
\(\sqrt{45^2 + (39 \div 2)^2}\) (= 49.043...) or \(\sqrt{15^2 + \left(\dfrac{39 \div 3}{2}\right)^2}\) (= 16.347...) oe | M1 |
\(\dfrac{2}{3} \times \text{``}{49.043...}\text{''}\) or \(2 \times \text{``}{16.347...}\text{''}\) (= 32.695...) oe OR \(\dfrac{1}{2} \times 13 \times \text{``}{16.347...}\text{''}\) (= 106.260...) or \(4 \times \dfrac{1}{2} \times 13 \times \text{``}{16.347...}\text{''}\) (= 425.042...) or \(\dfrac{1}{2} \times 39 \times \text{``}{49.043...}\text{''}\) (= 956.345...) or \(4 \times \dfrac{1}{2} \times 39 \times \text{``}{49.043...}\text{''}\) (= 3825.381...) | M1 |
\((39 \div 3)^2 + 39^2 + 4 \times \dfrac{(39 \div 3) + 39}{2} \times \text{``}{32.695...}\text{''}\) OR (39 ÷ 3)² + 39² + “3825.381...” – “425.042...” | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 5090 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for summing the area of the 2 squares – may be seen embedded in a calculation
M1: for finding the perpendicular slant height of either pyramid
M1: for finding the perpendicular slant height of the frustum OR the area of 1 or 4 triangular faces for either pyramid
M1: correct calculation for total surface area
A1: accept 5090 – 5091
M2 for eg \(\sqrt{\left(\dfrac{39 - 13}{2}\right)^2 + (45 - 15)^2}\) \((= \sqrt{1069} = 32.695...)\)