Higher November 2019 Paper 2 Q16
16.
(a) The volume of a cone is equal to the volume of a sphere.
The radius of the cone is four times the radius of the sphere.
The radius of the cone is four times the radius of the sphere.
Show that the perpendicular height of the cone is a quarter of the radius of the sphere. [4]
(b) The radii of two spheres are in the ratio 2 : 7.
The volume of the smaller sphere is 10·4 cm³.
The volume of the smaller sphere is 10·4 cm³.
Calculate the volume of the larger sphere. [3]
| Scheme | Marks | Comments |
|---|---|---|
| \(\tfrac13 \times \pi \times 16r^2 \times h = \tfrac43 \times \pi \times r^3\) | M2 | M1 for \(\tfrac13 \times \pi \times (4r)^2 \times h = \tfrac43 \times \pi \times r^3\), allow for \(\tfrac13 \times \pi \times 4r^2 \times h = \tfrac43 \times \pi \times r^3\) |
| \((h =)\ \dfrac{r}{4}\) | A2 | A1 for correct rearrangement but unsimplified form or \(r = 4h\), or FT from M1 to A1 for \((h =)\ r\) |
Notes
Alternative
| Scheme | Marks |
|---|---|
| \(\tfrac13 \times \pi \times 16r^2 \times r/4\) (or \(\tfrac13 \times \pi \times (4r)^2 \times r/4\) M1) | M2 |
| \(\tfrac43 \times \pi \times r^3\) (or A1 for 1 stage of simplification) | A2 |
| Scheme | Marks | Comments |
|---|---|---|
| Sight of 8 : 343 or \((7/2)^3\) or equivalent | B1 | |
| 445.9 (cm³) | B2 | B1 for \(3.5^3 \times 10.4\) or equivalent |
| (7) |