Two similar conical heaps of grain stand in a barn. The mass of a heap is proportional to the cube of its height. The smaller heap is 1.2 m high and has mas...
Question 1 Report
Two similar conical heaps of grain stand in a barn. The mass of a heap is proportional to the cube of its height. The smaller heap is 1.2 m high and has mass 540 kg.
- Find the mass of a heap of height 2 m. (2)
- Find the height of a heap of mass 1080 kg, correct to 3 significant figures. (2)
Answer Details
The figure shows two similar cones, one 1.2 m high and one 2 m high. Mass proportional to the cube of the height gives \(M = kh^3\).
Find the constant from the smaller heap: \(1.2^3 = 1.728\), so \(k = \frac{540}{1.728} = 312.5\) and \(M = 312.5h^3\).
(a)
- \(M = 312.5 \times 2^3 = 312.5 \times 8\). [1]
- \(= 2500\) kg. [1]
(b)
- \(1080 = 312.5h^3\), so \(h^3 = \frac{1080}{312.5} = 3.456\). [1]
- \(h = \sqrt[3]{3.456} = 1.5119\ldots\), so \(h = 1.51\) m to 3 significant figures. [1]
The cube law follows from similarity: because the heaps are the same shape, every length scales by the same factor, so the volume, and with constant density the mass, scales by the cube of that factor. Part (a) can be checked that way: \(\left(\frac{2}{1.2}\right)^3 = 4.6296\ldots\) and \(540 \times 4.6296\ldots = 2500\) kg.
Cube the height before multiplying by \(k\), and take the cube root, not the square root, when reversing the process in part (b).
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