Higher June 2023 Paper 2 Q11
11
| Volume of a sphere \(= \dfrac{4}{3}\pi r^3\) |
A bowl is a hemisphere with radius 12 cm
Water is poured into the bowl
at a rate of 325 cm\(^3\) per second
for 8 seconds.
Does the water fill more than 70% of the bowl?
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Compares 70% of volume of hemisphere with volume of water | ||
| \(\dfrac{4}{3} \times \pi \times 12^3\) or \(2304\pi\) or [7216, 7239.2] or \(\dfrac{2}{3} \times \pi \times 12^3\) or \(1152\pi\) or [3581, 3638] | M1 | oe eg \(\dfrac{4}{3}\pi \times 1728\) allow without any multiplication signs eg \(\dfrac{4}{3}\pi 12^3\) |
| \(0.7 \times\) their \(1152\pi\) or \(806.4\pi\) or [2506, 2547] | M1dep | oe \(0.7 \times\) their [3581, 3638] or \(\dfrac{4032}{5}\pi\) must be using volume of hemisphere |
| \(325 \times 8\) or 2600 | M1 | oe |
| [2506, 2547] and 2600 and Yes | A1 | oe |
| Alternative method 2 Works out volume of water as proportion of volume of hemisphere | ||
| \(\dfrac{4}{3} \times \pi \times 12^3\) or \(2304\pi\) or [7216, 7239.2] or \(\dfrac{2}{3} \times \pi \times 12^3\) or \(1152\pi\) or [3581, 3638] | M1 | oe eg \(\dfrac{4}{3}\pi \times 1728\) allow without any multiplication signs eg \(\dfrac{4}{3}\pi 12^3\) |
| \(325 \times 8\) or 2600 | M1 | oe |
| their \(2600 \div\) their \(1152\pi\) or [0.71, 0.73] | M1dep | oe eg their \(2600 \div\) their [3581, 3638] or 72% dep on M2 must be using volume of hemisphere |
| [71, 73](%) and Yes | A1 | oe eg 0.72 and 0.7 and Yes |
| Alternative method 3 Works out time to fill 70% of volume of hemisphere | ||
| \(\dfrac{4}{3} \times \pi \times 12^3\) or \(2304\pi\) or [7216, 7239.2] or \(\dfrac{2}{3} \times \pi \times 12^3\) or \(1152\pi\) or [3581, 3638] | M1 | oe eg \(\dfrac{4}{3}\pi \times 1728\) allow without any multiplication signs eg \(\dfrac{4}{3}\pi 12^3\) |
| \(0.7 \times\) their \(1152\pi\) or \(806.4\pi\) or [2506, 2547] or their \(1152\pi \div 325\) or [11, 11.2] | M1dep | oe \(0.7 \times\) their [3581, 3638] or \(\dfrac{4032}{5}\pi\) or their \([3581, 3638] \div 325\) must be using volume of hemisphere |
| \(0.7 \times\) their \(1152\pi \div 325\) or \(0.7 \times\) their \([3581, 3638] \div 325\) or [7.7, 7.84] | M1dep | oe their \([2506, 2547] \div 325\) or \(0.7 \times\) their [11, 11.2] |
| [7.7, 7.84] and Yes | A1 | oe |
Additional guidance
| Up to M3 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Allow 1.33(…) for \(\dfrac{4}{3}\) | |
| Allow 0.66(…) or 0.67 for \(\dfrac{2}{3}\) | |
| \(\pi\) may be seen as [3.14, 3.142] eg Alt 1 \(\dfrac{2}{3} \times 3.14 \times 12^3\) | M1 |
| If a number (or calculation) in terms of \(\pi\) is seen but \(\pi\) is subsequently omitted, treat as a miscopy for M marks eg Alt 1 | |
| \(1152\pi\) | M1 |
| \(0.7 \times 1152 = 806.4\) | M1dep |
| \(325 \times 8 = 2600\) Yes | M1A0 |
| Yes cannot be implied by inequalities | |
| Alts 1 and 2 325 cm\(^3 \times 8\) seen is M1 even if evaluated incorrectly \(325^3 \times 8\) seen is M0 unless recovered to 2600 | |
| Do not allow misreads of the given formula unless recovered eg1 using \(12^2\) instead of \(12^3\) eg2 using \(\dfrac{3}{4}\) instead of \(\dfrac{4}{3}\) | |
| For \(0.7 \times\) their \(1152\pi\), do not accept \(70\% \times\) their \(1152\pi\) unless recovered |