Foundation November 2021 Paper 2 Q27

AQACurrent spec4 marksVolume & Surface Area

27 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]

Mark scheme

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

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 \(5000 \div 88.35\ldots\)	M1M1M0
Yes can be implied eg Alt 1 \(57 < 60\)	M3A1