Numeracy Higher June 2022 Unit 2 Q9
9. SureCan is a company that makes cylindrical cans for the food industry.
Two of the cans it makes are mathematically similar.
They are shown below.
Diagrams not drawn to scale
The base area of the large can is 2·25 times the base area of the small can.
(a) Show that the height of the small can is 8 cm. [3]
(b) The volume of the small can is 144 cm3.
Calculate the base area of the large can. [2]
Calculate the base area of the large can. [2]
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\sqrt{2.25}\) OR Area scale factor = 1.52 OR Area scale factor = 2.25 AND scale factor = 1.5 | M1 | |
| (Height =) 12 ÷ \(\sqrt{2.25}\) or 12 ÷ 1.5 or 12 × 2/3 | m1 | |
| = 8 (cm) | A1 | Must be from convincing working |
| Scheme | Marks | Comments |
|---|---|---|
| (Base area of large can =) 144 ÷ 8 × 2.25 or 18 × 2.25 | M1 | Note: 2.25 could be written as 1.52 |
| = 40.5 (cm2) | A1 |
Alternative
9(b) Alternative method 1:
| Scheme | Marks | Comments |
|---|---|---|
| (Base area of large can =) \(144 \times \sqrt{2.25}^{\,3} \div 12\) or 144 × 1.53 ÷ 12 | M1 | |
| = 40.5 (cm2) | A1 |
Alternative
9(b) Alternative method 2:
| Scheme | Marks | Comments |
|---|---|---|
| (Radius of large can =) \(\sqrt{\dfrac{144}{8 \times \pi}} \times \sqrt{2.25}\) or \(\sqrt{\dfrac{18}{\pi}} \times 1.5\) (=3.59 to 3.592) | M1 | |
| (Base area of large can =) 40.5 (cm2) | A1 | From \(\boldsymbol{\pi} \times \left(\sqrt{\dfrac{18}{\pi}} \times 1.5\right)^2\) |