Higher June 2019 Paper 1 Q21
21 A solid is made from a hemisphere and a cylinder.
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
Diagram NOT accurately drawn
The hemisphere and the cylinder have the same radius.
The ratio of the radius of the cylinder to the height of the cylinder is \(1 : 3\)
Given that the solid has volume \(792\pi\) cm3
work out the height of the solid.
(5)
| Scheme | Marks |
|---|---|
| \(h = 3r\) or \(r = \dfrac{h}{3}\) | M1 |
| \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi r^3\) oe or \(\pi \times r^2 \times 3r\) oe | M1 |
| \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi r^3 + \pi \times r^2 \times 3r = 792\pi\) oe | M1 |
| (r =) 6 or (h =) 18 | A1 |
| 24 | A1ft |
| (5) | |
| (5 marks) |
Notes
M1: for \(h = 3r\) or \(r = \dfrac{h}{3}\) oe stated or used correctly
M1: or \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi\left(\dfrac{h}{3}\right)^3\) or \(\pi\left(\dfrac{h}{3}\right)^2 h\)
M1: or
\(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi\left(\dfrac{h}{3}\right)^3 + \pi\left(\dfrac{h}{3}\right)^2 h = 792\pi\)
A1ft: their “6” × 4 or \(\text{``}{18}\text{''} \times \dfrac{4}{3}\) correctly evaluated dep on M3