Higher June 2022 Paper 2 Q13
13. A cuboid of copper has a mass of 2150·4 grams and a volume of 240 cm3.
A sphere of copper has a radius of \(x\) cm.
Show that the mass, in grams, of the sphere is less than \(38x^3\). [4]
| Scheme | Marks | Comments |
|---|---|---|
| If a mixture of methods is employed, use the one that awards marks to the candidates advantage. | ||
| (Density of copper =) \(\dfrac{2150.4}{240}\) | M1 | |
| 8.96 or 9 (g/cm3) | A1 | |
| (mass of sphere =) \(\dfrac43 \pi x^3 \times 8.96\) | m1 | FT ‘their 8.96’ |
| \(37(.5\ldots..)x^3\ \ (\lt 38x^3)\) | A1 | Allow answers in the range \(37(x^3)\) to \(37.7(x^3)\) from correct working. |
Alternative method 1
| Scheme | Marks | Comments |
|---|---|---|
| (For equal volumes) \(x^3 = 240 \times \frac34 \div \pi\) \(= 57.29\ldots..\) | M1 A1 | Allow for \(240 = \frac43 \times \pi \times x^3\) Allow answers in the range 57.28 to 57.9 May be seen in later working |
| EITHER (For equal mass multiplier of \(x^3\) =) \(2150 \div 57.29\) | m1 | |
| = 37.53…(and less than 38) | A1 | |
| Alternative method for the last two marks OR (If multiplier of \(x^3 = 38\), mass =) \(38 \times 57.29\ldots.\) | m1 | |
| = 2177(.2…...g.) AND this is greater than 2150.4g | A1 |
Alternative method 2
| Scheme | Marks | Comments |
|---|---|---|
| (Density of copper =) \(\dfrac{2150.4}{240}\) | M1 | |
| 8.96 or 9 (g/cm3) | A1 | |
| \(\dfrac43 \pi \times k = 38\) si | M1 | |
| \(k = 9.07\ldots.\) (and it is less than this so mass is less than \(38x^3\)) | A1 |
| (4) |