Higher June 2022 Paper 1 Q22
22 A solid is made from a cone and a hemisphere.
Diagram NOT accurately drawn
The circular plane face of the hemisphere coincides with the circular base of the cone.
The radius of the hemisphere and the radius of the circular base of the cone are both 20 cm.
The curved surface area of the cone is \(580\pi\) cm2
The volume of the solid is \(k\pi\) cm3
Work out the exact value of \(k\)
(5)
| Scheme | Marks |
|---|---|
| \(580\pi = \pi \times 20 \times l\) oe | M1 |
| \((l =)\;\dfrac{580\pi}{20\pi}\;(= 29)\) | M1 |
| \(\sqrt{\text{``}{29}\text{''}^2 - 20^2}\;\left(= \sqrt{441} = 21\right)\) | M1 |
\(\left(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times 20^3\right) + \left(\dfrac{1}{3} \times \pi \times 20^2 \times \text{``}{21}\text{''}\right)\) or \(\dfrac{16\,000}{3}\pi + \dfrac{8400}{3}\pi\) or \(\dfrac{16\,000}{3}\pi + 2800\pi\) | M1 |
| \(\dfrac{24\,400}{3}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for correct substitution into \(A = \pi rl\)
M1: for a complete method
(Award M4 for 8133.3…….. if \(\dfrac{24\,400}{3}\) is not seen)
A1: \(8133.\dot{3}\) or \(8133\dfrac{1}{3}\) (as exact form was requested)
SC B4 for an answer of 25551(.62….) if no method shown