4MA0 Higher June 2012 Paper 4 Q22
22
Diagram NOT accurately drawn
Two solid spheres, each of radius \(r\) cm, fit exactly inside a hollow cylinder.
The radius of the cylinder is \(r\) cm.
The height of the cylinder is equal to \(4r\) cm.
The volume of the space inside the cylinder, not occupied by the spheres, is \(\dfrac{125}{6}\pi\) cm\(^3\)
Calculate the value of \(r\).
Show your working clearly.
(5)
| Scheme | Marks |
|---|---|
| \(\pi r^2 \times 4r - 2 \times 4\pi r^3/3 = 125\pi/6\) oe \(24r^3 - 16r^3 = 125\) oe | M2 |
| \(r^3 = 125/8\) oe | M1 |
| \(r = \sqrt[3]{(125/8)}\) | M1 |
| 2.5 | A1 |
| (Total 5 marks) |
Notes
M2: Any equation based on cylinder – 2 spheres = space oe
\(h = 4r\) must be implicit for award of M2
{decimal form: \(12.6r^3 - 8.4r^3 = 65.4\) (1 dp or better)}
If not M2 then M1 for \(\pi r^2 \times 4r\) or better
M1: One occurrence of \(r^3\) in correct equation.
A1: awrt to 2.5 Ans dep on M3