1MA0 Higher June 2015 Paper 1 Q23
23 The diagram shows a container for grain.
Diagram NOT accurately drawn
The container is a cylinder on top of a cone.
The cylinder has a radius of 3 m and a height of \(h\) m.
The cone has a base radius of 3 m and a vertical height of 4 m.
The container is empty.
The container is then filled with grain at a constant rate.
After 5 hours the depth of the grain is 6 metres above the vertex of the cone.
After 9 hours the container is full of grain.
Work out the value of \(h\).
Give your answer as a fraction in its simplest form.
You must show all your working. (5)
| Answer | Mark | Notes |
|---|---|---|
| \(\dfrac{14}{3}\) | 5 | M1 for correct substitution into a volume formula for a cylinder or a cone, eg. \(\tfrac{1}{3} \times \pi \times 3^2 \times 4\) \((= 12\pi)\) or \(\pi \times 3^2 \times (6 - 4)\) \((= 18\pi)\) or \(\pi \times 3^2 h\) \((= 9\pi h)\) or \(\pi \times 3^2 (h - 2)\) M1 for method to find volume after 5 hours, eg. \(\text{``}12\pi\text{''} + \text{``}18\pi\text{''}\) \((= 30\pi)\) M1 (dep on M1) for use of a correct ratio, eg. \(\text{``}30\pi\text{''} \times \tfrac{9}{5}\) \((= 54\pi)\) or \(\text{``}30\pi\text{''} \times \tfrac{4}{5}\) \((= 24\pi)\) M1 for deriving an equation in \(h\), eg. \(\text{``}54\pi\text{''} = \text{``}9\pi h\text{''} + \text{``}12\pi\text{''}\) A1 for \(\dfrac{14}{3}\) or equivalent fraction |