Foundation June 2018 Paper 2 Q27
27 Volume of a sphere \(= \dfrac{4}{3}\pi r^3\) where \(r\) is the radius
A container is a hemisphere of radius 30 cm
Sand fills the container at a rate of 4000 cm\(^3\) per minute.
Does it take less than a quarter of an hour to fill the container?
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{4}{3}\pi \times 30^3\) or \(36\,000\pi\) or [112 757, 113 112] or \(\dfrac{1}{2} \times \dfrac{4}{3}\pi \times 30^3\) or \(18\,000\pi\) or [55 954, 56 839] | M1 | oe allow 1.33… for \(\dfrac{4}{3}\) allow 0.66… or 0.67 for \(\dfrac{2}{3}\) |
| their [112 757, 113 112] \(\div\) 4000 or \(9\pi\) or 28.(…) or their [55 954, 56 839] \(\div\) 4000 or \(\dfrac{9\pi}{2}\) or [13.9, 14.21] or their [112 757, 113 112] \(\div\) (4000 \(\times\) 60) or \(\dfrac{3\pi}{20}\) or [0.46, 0.4713] or their [55 954, 56 839] \(\div\) (4000 \(\times\) 60) or \(\dfrac{3\pi}{40}\) or 0.23… or 0.24 | M1dep | |
| [13.9, 14.21] and Yes or 0.23… or 0.24 and Yes | A1 | |
| Alternative method 2 | ||
| \(\dfrac{4}{3}\pi \times 30^3\) or \(36\,000\pi\) or [112 757, 113 112] or \(\dfrac{1}{2} \times \dfrac{4}{3}\pi \times 30^3\) or \(18\,000\pi\) or [55 954, 56 839] | M1 | oe allow 1.33… for \(\dfrac{4}{3}\) allow 0.66… or 0.67 for \(\dfrac{2}{3}\) |
| \(4000 \times 15\) or 60 000 | M1 | |
| [55 954, 56 839] and 60 000 and Yes | A1 | |
| Alternative method 3 | ||
| \(\dfrac{4}{3}\pi \times 30^3\) or \(36\,000\pi\) or [112 757, 113 112] or \(\dfrac{1}{2} \times \dfrac{4}{3}\pi \times 30^3\) or \(18\,000\pi\) or [55 954, 56 839] | M1 | oe allow 1.33… for \(\dfrac{4}{3}\) allow 0.66… or 0.67 for \(\dfrac{2}{3}\) |
| their [112 757, 113 112] \(\div\) 15 or \(2400\pi\) or [7517, 7541] or their [55 954, 56 839] \(\div\) 15 or \(1200\pi\) or [3730, 3790] | M1dep | |
| [3730, 3790] and Yes | A1 | |
Additional guidance
Do not award A1 if incorrect conversion of \(\dfrac{1}{4}\) hour seen