14 The diagram shows a square-based pyramid ABCDE. O is the centre of the base.
The pyramid has base length 20 cm and each sloping edge has length 14.5 cm.
(a) Draw the plan view of the pyramid on the one-centimetre grid below.
Scale: 1 cm represents 4 cm.
[2]
(b) Calculate the volume of the pyramid. You must show your working.
[The volume of a pyramid is \(\frac{1}{3}\) × area of base × perpendicular height] [5]
Mark scheme (a)
Answer
Marks
Part marks and guidance
with side 5 cm
2
B1 for a square drawn with side 5 cm or for a square of any length with two diagonals
Mark intention 2mm tolerance radially on centre point by eye
Mark scheme (b)
Answer
Marks
Part marks and guidance
\(\frac{200\sqrt{41}}{3}\) or 426.6 to 427 with correct working
5
M3 for \(\sqrt{14.5^2 - 10^2 - 10^2}\) oe
or M2 for \(14.5^2 - 10^2 - 10^2\) oe or M1 for \(14.5^2 = 10^2 + 10^2 + h^2\) or for \(10^2 + 10^2\) oe implied by 200 or for \(\sqrt{10^2 + 10^2}\) oe implied by 14.1…or \(10\sqrt{2}\) or for \(20^2 + 20^2\) oe implied by 800 or for \(\sqrt{20^2 + 20^2}\) oe implied by 28.2[8..], 28.3 …or \(20\sqrt{2}\)
AND
M1 for \(\frac{1}{3} \times 20 \times 20 \times their\ 3.2[0]\) or for \(\frac{1}{3} \times 400 \times their\ 3.2[0]\)
If 0 or 1 scored, instead award SC2 for \(\frac{200\sqrt{41}}{3}\) or 426.6 to 427 with no or insufficient working
If 0 scored, SC1 for \(\frac{\sqrt{41}}{2}\) or \(\sqrt{10.25}\) or 3.2[0] to 3.211 with no or insufficient working
‘Correct working’ requires evidence of at least three M marks Working may be on diagram May be seen in stages Method must be seen for M3 (3.2[0] to 3.211 or \(\frac{\sqrt{41}}{2}\) or \(\sqrt{10.25}\) ) (10.25)
May be seen in stages
For M3 and M2 condone \(10\sqrt{2}{}^2\) for \((10\sqrt{2})^2\) which may lead to wrong answer for M2 of 190.25 or for M3 of 13.79…
Their 3.2[0] must be in the range 3.2[0] to 3.211 and must come from an attempt at 3D trig or 3D Pythagoras
