Higher January 2021 Paper 1 Q17
17 A solid, S, is made from a hemisphere and a cylinder.
The centre of the circular face of the hemisphere and the centre of the top face of the cylinder are at the same point.
Diagram NOT accurately drawn
The radius of the cylinder and the radius of the hemisphere are both \(x\) cm.
The height of the cylinder is \((20 - 4x)\) cm.
The volume of S is \(V\) cm3 where \(V = \dfrac{1}{3}\pi y\)
Find the maximum value of \(y\).
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
e.g. \((V =)\;\dfrac{1}{2}\left(\dfrac{4}{3}\pi x^3\right) + \pi x^2(20 - 4x)\) or \((V =)\;\dfrac{2}{3}\pi x^3 + 20\pi x^2 - 4\pi x^3\) | M1 |
e.g. \(\dfrac{1}{3}\pi y = \dfrac{1}{2}\left(\dfrac{4}{3}\pi x^3\right) + \pi x^2(20 - 4x)\) or \(\dfrac{1}{3}\pi y = \dfrac{2}{3}\pi x^3 + 20\pi x^2 - 4\pi x^3\) | M1 |
| \(y = 60x^2 - 10x^3\) oe | A1 |
| e.g. \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x} =\right) 120x - 30x^2 = 0\) oe | M1 |
| Working required Answer: 320 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for a correct expression
M1: for a correct equation
A1: for writing \(y\) in terms of \(x\)
M1: for differentiating their \(ax^2 + bx^3\) and equating to 0
A1: (dep on M3) cao