Higher November 2021 Paper 2 Q9
9 A ball contains 5000 cm3 of air.
More air is pumped into the ball at a rate of 160 cm3 per second.
The ball is full of air when it becomes a sphere with radius 15 cm
Volume of a sphere = \(\dfrac{4}{3}\pi r^3 \quad\) where \(r\) is the radius
Does it take less than 1 minute to fill the ball?
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Working out time to fill the ball | ||
| \(4 \div 3 \times 15^3 \times \pi\) or \([4488, 4500]\pi\) or \([14\,092, 14\,139]\) | M1 | oe allow 1.33 or better |
| their \([14\,092, 14\,139] - 5000\) or \([9092, 9139]\) or their \([14\,092, 14\,139] \div 160\) or \([88, 88.37]\) | M1dep | oe |
| (their \([14\,092, 14\,139] - 5000) \div 160\) or \([56, 57.12]\) | M1dep | oe eg their \([9092, 9139] \div 160\) or their \([88, 88.37] - 5000 \div 160\) |
| \([56, 57.12]\) and Yes | A1 | |
| Alternative method 2 Comparing volume needed with volume that could be filled | ||
| \(4 \div 3 \times 15^3 \times \pi\) or \([4488, 4500]\pi\) or \([14\,092, 14\,139]\) | M1 | oe allow 1.33 or better |
| their \([14\,092, 14\,139] - 5000\) or \([9092, 9139]\) | M1dep | |
| \([58, 60] \times 160\) or \([9280, 9600]\) | M1 | oe |
| \([9092, 9139]\) and \([9280, 9600]\) and Yes | A1 | |
| Alternative method 3 Volume of ball compared with volume that could be filled + 5000 | ||
| \(4 \div 3 \times 15^3 \times \pi\) or \([4488, 4500]\pi\) or \([14\,092, 14\,139]\) | M1 | oe allow 1.33 or better |
| \([58, 60] \times 160\) or \([9280, 9600]\) | M1 | oe |
| their \([9280, 9600] + 5000\) or \([14\,280, 14\,600]\) | M1dep | dep on 2nd M1 |
| \([14\,092, 14\,139]\) and \([14\,280, 14\,600]\) and Yes | A1 | |
Additional guidance
| Accept \(\dfrac{4}{3}\pi 15^3\) without multiplication signs | |
| Condone use of 1.3 for up to M3 if 1.3 shown | |
| Up to M3 may be awarded for correct work, with no or incorrect answer, even if this is seen amongst multiple attempts | |
| Using an incorrect power eg \(15^2\), \(15\pi^3\), \((15\pi)^3\) or omitting \(\pi\) unless recovered | 1st M0 |
| NB 56.(59…) or 56.6 or 57 coming from \(5000 \div 88.35…\) | M1M1M0 |
| Yes can be implied eg Alt 1 \(57 \lt 60\) | M3A1 |