4MA0 Higher June 2011 Paper 3 Q23
23
Diagram NOT accurately drawn
The diagram shows a solid cylinder and a solid sphere.
The cylinder has radius \(r\).
The sphere has radius \(r\).
Given that \(\dfrac{\textbf{Total}\text{ surface area of cylinder}}{\text{Surface area of sphere}} = 2\)
find the value of \(\dfrac{\text{Volume of cylinder}}{\text{Volume of sphere}}\)
(5)
| Scheme | Marks |
|---|---|
| \(\dfrac{2\pi r^2 + 2\pi rh}{4\pi r^2} = 2\) | M1 |
| \(2\pi r^2 + 2\pi rh = 2 \times 4\pi r^2\) oe | M1 |
| \(h = 3r\) oe | A1 |
| \(\dfrac{\pi r^2 \times \text{“}3r\text{”}}{\frac{4}{3}\pi r^3}\) oe | M1 |
| \(\frac{9}{4}\) oe | A1 |
| (Total 5 marks) |
Notes
M1: Also award for \(\dfrac{\pi r^2 + 2\pi rh}{4\pi r^2} = 2\)
M1: for \(2\pi r^2 + 2\pi rh = 2 \times 4\pi r^2\) oe
or \(\dfrac{2\pi r(r + h)}{4\pi r^2} = 2\)
If first M1 awarded for \(\dfrac{\pi r^2 + 2\pi rh}{4\pi r^2} = 2\) award this second M1 also for \(\pi r^2 + 2\pi rh = 2 \times 4\pi r^2\) oe
A1: If first M1 awarded for \(\dfrac{\pi r^2 + 2\pi rh}{4\pi r^2} = 2\) and second M1 for \(\pi r^2 + 2\pi rh = 2 \times 4\pi r^2\) oe
Award this A1 also for \(h = 3.5r\) oe
M1: dep on first two M1s
\(h\) must be of the form \(kr\)