Numeracy Higher November 2022 Unit 1 Q9
9.
On a sunny Saturday, Gerallt sold 400 ice creams in total in 4 hours in the 2 shops.
Gerallt is considering opening another ice cream shop.
He expects the sales per hour in each of the 3 shops to be the same.
His ice cream shops will be open for 5 hours a day.
Calculate how many ice creams Gerallt would expect to sell in total on a sunny Saturday in 5 hours in the 3 shops.
You must show all your working. [3]
He has designed his new ice cream shop to look like half an ice cream cone.
The design consists of half a hollow cone connected to half a hollow hemisphere, as shown below.
Diagrams not drawn to scale
The radius of the base of the half hemisphere is 3 m.
The perpendicular height of the cone is shown as \(h\) on the diagram above.
Gerallt designed the shop so that the volume of the half cone is equal to the volume of the half hemisphere.
Calculate the length \(x\). Give your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are both integers and \(b\) is as small as possible. [8]
From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\); Volume of cone \(= \dfrac{1}{3}\pi r^2 h\)
| Scheme | Marks | Comments | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(400 \times \dfrac{5}{4} \times \dfrac{3}{2}\) OR 400 × 1.25 × 1.5 OR 50 × 5 × 3 | M2 | A table method altering all 3 at the same time is M0 M1 for 400 × 5/4 (= 500) or 400 × 3/2 (= 600) or equivalent e.g.
| ||||||||||||||||
| = 750 (ice creams) | A1 | CAO |
| Scheme | Marks | Comments |
|---|---|---|
| (Volume of half cone =) \(\dfrac{1}{6}\) × π × 32 × h | B1 | or equivalent e.g. \(\dfrac{9\pi}{6}\) × h |
| (Volume of half hemisphere =) \(\dfrac{1}{3}\) × π × 33 | B1 | or equivalent e.g. 9π |
| \(\dfrac{1}{6}\) × π × 32 × h = \(\dfrac{1}{3}\) × π × 33 or equivalent | M1 | FT their volumes provided B1 previously awarded and the incorrect volume is a multiple of the correct one Note: An equation using consistent multiples of these expressions that, when correctly solved, would give the answer of 6(m) implies the previous B1 marks e.g. a cone equated to a hemisphere, \(\frac{1}{3}\pi \times 3^{2} \times h = \frac{2}{3}\pi \times 3^{3}\) is awarded B1B1M1 and possible A1, but a cone equated to a sphere, \(\frac{1}{3}\pi \times 3^{2} \times h = \frac{4}{3}\pi \times 3^{3}\) is awarded B0B0M0A0 |
| h = 6 (m) | A1 | Allow an unsupported answer of h = 6 (m) for B1B1M1A1 May be seen in further working |
| (x2 =) 62 + 32 | M1 | FT ‘their derived 6’ |
| x2 = 45 or (x =) \(\sqrt{45}\) | A1 | |
| (x =) 3\(\sqrt{5}\) (m) | B2 | For B2, FT ‘their derived 45 (their x, not their h)’ provided their surd can be written in the form \(a\sqrt{b}\) B1 for writing 45 as a product of 2 or more factors where one of the factors OR the product of a pair of their factors is a square number e.g. 9 × 5, 3 × 3 × 5, OR B1 for writing √45 as a product of 2 or more factors where one of the factors OR the product of a pair of their factors gives a whole number e.g. √3 × √3 × √5, √9 × √5 |