Higher June 2022 Paper 2 Q13
13 Outside a cafe there is a large plastic ice cream cornet.
The cornet is a hemisphere on top of a cone.
The cone and the hemisphere each have radius 24 cm
The cone has perpendicular height 117 cm
| Volume of a cone \(= \dfrac{1}{3}\pi r^2 h\) \(r\) is the radius \(h\) is the perpendicular height | Volume of a hemisphere \(= \dfrac{2}{3}\pi r^3\) \(r\) is the radius |
(a) Work out the total volume of the cornet. [4 marks]
(b) The actual cornets that the cafe sells are similar to the plastic one.
For the actual cornets, the cone and the hemisphere each have radius 2 cm
How many times greater is the volume of the plastic cornet than an actual cornet? [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{1}{3} \times \pi \times 24^2 \times 117\) or \(\dfrac{2}{3} \times \pi \times 24^3\) | M1 | oe eg \(\dfrac{1}{3}\pi \times 576 \times 117\) or \(\dfrac{2}{3}\pi \times 13\,824\) |
| \(22\,464\pi\) or [70 536, 70 582] or \(9216\pi\) or [28 938, 28 957] | A1 | may be seen in a sum implied by final A1 |
| \(\dfrac{1}{3} \times \pi \times 24^2 \times 117 + \dfrac{2}{3} \times \pi \times 24^3\) or \(22\,464\pi + 9216\pi\) or [70 536, 70 582] + [28 938, 28 957] | M1dep | oe |
| \(31\,680\pi\) or [99 474, 99 539] | A1 |
Additional guidance
| \(\pi\) may be seen as any value in the interval [3.14, 3.142] | |
| Do not allow any misreads of formulae unless recovered eg \(\pi \times 24^2 \times 117\) and \(\dfrac{2}{3} \times \pi \times 24^2\) | M0 |
| Allow dots for multiplication | |
| For A marks allow eg \(22\,464 \times \pi\) or \(\pi \times 31\,680\) | |
| \(31\,680\pi\) followed by incorrect evaluation attempt | M1A1M1A1 |
| \(31\,680\pi\) followed by further work | M1A1M1A0 |
| 31 680 only | M0A0M0A0 |
| \(\dfrac{1}{3} \times \pi \times 24^2 \times 117 = 4725 \qquad \dfrac{2}{3} \times \pi \times 24^3 = 28\,952\) \(4725 + 28\,952\) (even though 4725 is wrong the method for \(\dfrac{1}{3} \times \pi \times 24^2 \times 117\) is seen) | M1A1 M1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Uses volume scale factor | ||
| \(24 \div 2\) or 12 | M1 | oe eg \(12 \times 2 = 24\) |
| (their \(12)^3\) | M1dep | oe eg \(24^3 \div 2^3\) or \(13\,824 \div 8\) |
| 1728 | A1 | condone 1 : 1728 or 1728 : 1 SC2 \(\dfrac{1}{1728}\) |
| Alternative method 2 Compares volumes of cornets (ie compares total volumes) | ||
| \(24 \div 2\) or 12 | M1 | oe eg \(12 \times 2 = 24\) may be implied eg (height of cone) 9.75 or (volume of cone) \(13\pi\) or (volume of cone) [40.8, 40.85] or (total volume) \(\dfrac{55}{3}\pi\) or [57.4, 57.7] |
| their (a) \(\div\) \(\left(\dfrac{1}{3}\pi \times 2^2 \times \dfrac{117}{\text{their } 12} + \dfrac{2}{3}\pi \times 2^3\right)\) | M1dep | oe eg their (a) \(\div\) [57.4, 57.7] |
| 1728 | A1 | condone 1 : 1728 or 1728 : 1 SC2 \(\dfrac{1}{1728}\) |
| Alternative method 3 Compares volumes of cones | ||
| \(24 \div 2\) or 12 | M1 | oe eg \(12 \times 2 = 24\) may be implied eg (height of cone) 9.75 or (volume of cone) \(13\pi\) or (volume of cone) [40.8, 40.85] or (total volume) \(\dfrac{55}{3}\pi\) or [57.4, 57.7] |
| their volume of cone from (a) \(\div \left(\dfrac{1}{3}\pi \times 2^2 \times \dfrac{117}{\text{their } 12}\right)\) | M1dep | oe eg their volume of cone from (a) \(\div\) [40.8, 40.85] |
| 1728 | A1 | condone 1 : 1728 or 1728 : 1 SC2 \(\dfrac{1}{1728}\) |
| Alternative method 4 Compares volumes of hemispheres | ||
| their volume of hemisphere from (a) \(\div \left(\dfrac{2}{3}\pi \times 2^3\right)\) | M2 | oe eg their volume of hemisphere from (a) \(\div\) [16.7, 16.8] |
| 1728 | A1 | condone 1 : 1728 or 1728 : 1 SC2 \(\dfrac{1}{1728}\) |
Additional guidance
| \(\pi\) may be seen as any value in the interval [3.14, 3.142] | |
| Answer \(\times 1728\) or \(1728 \times\) | M1M1A1 |
| Answer 12 | M1M0A0 |
| Answer \(12^3\) with 1728 seen | M1M1A1 |
| Answer \(12^3\) without 1728 seen | M1M1A0 |
| Alts 2, 3 and 4 Allow if an incorrect volume formula from (a) is used in (b) eg Alt 4 (a) \(\dfrac{1}{2} \times \dfrac{2}{3} \times \pi \times 24^3 = 4608\pi\) (b) \(\dfrac{1}{2} \times \dfrac{2}{3} \times \pi \times 2^3 = \dfrac{8}{3}\pi\) \(4608\pi \div \dfrac{8}{3}\pi\) 1728 | M2 A1 |
| Alts 2 and 3 Allow \(\dfrac{55}{3}\) rounded to 1dp or better eg allow 18.3 | |
| Alt 4 Allow \(\dfrac{16}{3}\) rounded to 1dp or better eg allow 5.3 | |
| Alts 2 and 3 2nd M1 – allow consistent omission of \(\pi\) | |
| Alt 4 M2 – allow consistent omission of \(\pi\) | |
| Alts 2, 3 and 4 Answer 1728 is M1M1A1 unless it comes from rounding or truncating | |
| eg1 Alt 2 \(99\,525.655 \div 57.595 = 1728\) | M1M1A1 |
| eg2 Alt 2 \(99\,525.655 \div 57.595 = 1728.03\) Answer 1728 | M1M1A0 |