Higher November 2021 Paper 2 Q15
15. The diagram shows a candle. It is made by removing a small pyramid of wax from the top of a large pyramid of wax, to expose the wick.
Diagram not drawn to scale
The perpendicular height of the large pyramid is 24 cm.
The base of the large pyramid is a square with sides 12 cm.
The base of the small pyramid that is removed is a square with sides 2 cm.
The density of the candle wax is 0·9 g/cm\(^3\).
Volume of pyramid = \(\dfrac{1}{3} \times\) area of base \(\times\) perpendicular height
(a) Calculate the mass of the candle.
State one assumption you make. [7]
State one assumption you make. [7]
(b) Describe the impact of your assumption on your answer to part (a). [1]
| Scheme | Marks | Comments |
|---|---|---|
| Mark using one method only, to the candidate’s advantage | ||
| (Height of pyramid removed = ) 4 (cm) | B1 | |
| \(\frac{1}{3} \times 24 \times 12^2 - \frac{1}{3} \times 4 \times 2^2\) (= 1152 – 5.33…) | M2 | FT ‘their 4’ provided < 24 M1 for \(\frac{1}{3} \times 24 \times 12^2\) or \(\frac{1}{3} \times 4 \times 2^2\) |
| \(\dfrac{3440}{3}\) or 1146.6(66…) or 1146.7 | A1 | FT ‘their 4’ Allow 1147 from correct working |
| (Mass = ) \(0.9 \times \dfrac{3440}{3}\) | M1 | FT ‘their \(\dfrac{3440}{3}\)’ |
| 1032 (grams) | A1 | CAO Award for 1031.9(4) to 1032.3 |
| Valid assumption e.g. ‘No wax is lost when the wax is cut.’ or ‘The mass of the wick is negligible.’ or ‘The wick can be ignored.’ or ‘The volume of the wick is negligible.’ or ‘The wick has the same density as the wax.’ | E1 |
Alternative Method 1 for first 6 marks (difference of masses)
| Scheme | Marks | Comments |
|---|---|---|
| (Height of pyramid removed = ) 4 cm | B1 | |
| \(\frac{1}{3} \times 24 \times 12^2\) or \(\frac{1}{3} \times 4 \times 2^2\) | M1 | (1152 or 5.33…) ‘FT their 4’ |
| \(0.9 \times \frac{1}{3} \times 24 \times 12^2\) and \(0.9 \times \frac{1}{3} \times 4 \times 2^2\) | M2 | FT ‘their 4’ for M2 or M1 M1 for \(0.9 \times \frac{1}{3} \times 24 \times 12^2\) or \(0.9 \times \frac{1}{3} \times 4 \times 2^2\); (1036.8 or 1037) (4.77 or 4.8) |
| \(\left(0.9 \times \frac{1}{3} \times 24 \times 12^2\right) - \left(0.9 \times \frac{1}{3} \times 4 \times 2^2\right)\) | M1 | |
| 1032 (grams) | A1 | CAO Award for 1031.9(4) to 1032.3 |
Alternative Method 2 for first 6 marks (scaling the volume)
| Scheme | Marks | Comments |
|---|---|---|
| \(\frac{1}{3} \times 24 \times 12^2\) | M1 | (=1152) |
| \(0.9 \times \frac{1}{3} \times 24 \times 12^2\) or \(\dfrac{1}{6^3}\left(\frac{1}{3} \times 24 \times 12^2\right)\) | M1 | (1036.8 or 1037) or (4.77 or 4.8) |
| \(0.9\left(\frac{1}{3} \times 24 \times 12^2 - \dfrac{1}{6^3}\left(\frac{1}{3} \times 24 \times 12^2\right)\right)\) oe | M3 | or M2 for sight of \(\frac{1}{3} \times 24 \times 12^2 - \dfrac{1}{6^3}\left(\frac{1}{3} \times 24 \times 12^2\right)\) oe or M1 for sight of \(\dfrac{1}{6^3}\left(0.9 \times \frac{1}{3} \times 24 \times 12^2\right)\) oe |
| 1032 (grams) | A1 | CAO Award for 1031.9(4) to 1032.3 |
| Scheme | Marks | Comments |
|---|---|---|
| Valid impact following their assumption e.g. ‘The volume of the wax may be a little less so the mass will be less.’ or ‘The density of the wick is likely to be less than the density of the wax, so in fact the mass of the candle is likely to be a bit less.’ | E1 | Must have stated an assumption in part (a). |
| (8) |