Grain is tipped from a lorry onto level ground on a farm, forming a conical heap. The heap has a base radius of 10 m and a volume of \(1000\) m\(^3\). The f...
Question 1 Report
Grain is tipped from a lorry onto level ground on a farm, forming a conical heap. The heap has a base radius of 10 m and a volume of \(1000\) m\(^3\). The farmer wants to know how tall the heap has grown before it starts to topple.
- Work out the height of the heap, correct to 3 significant figures. (3)
Answer Details
The conical heap's volume formula links its base radius, height and volume, so when the volume and radius are known, the formula is rearranged to find the missing height.
Substituting the known volume and radius into \(V = \frac{1}{3}\pi r^2 h\):
\[\frac{1}{3}\pi \times 10^2 \times h = 1000 \quad \textbf{[1 mark]}\]
Rearranging to make \(h\) the subject:
\[h = \frac{3000}{100\pi} \quad \textbf{[1 mark]}\]
\[h = 9.55 \text{ m (3 s.f.)} \quad \textbf{[1 mark]}\]
Exam tip: when rearranging \(\frac{1}{3}\pi r^2 h = V\) for \(h\), multiply both sides by 3 first to clear the fraction, then divide by \(\pi r^2\); doing these steps in a different order is more likely to introduce an arithmetic error.
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