Higher June 2018 Paper 3 Q21

21 The mass of an ornament is \(m\) grams.

The height of the ornament is \(h\) centimetres.

\(m\) is directly proportional to the cube of \(h\).

\(m = 1600\) when \(\;h = 8\)

(a) Work out an equation connecting \(m\) and \(h\). [3 marks]

(b) Work out the mass of an ornament of height 12 centimetres. [2 marks]

Mark scheme (a)

Answer	Mark	Comments
\(m \propto h^3\) or \(m = \mathrm{k} \times h^3\) or \(1600 = \mathrm{k} \times 8^3\) or \(\mathrm{c} \times m = h^3\) or \(\mathrm{c} \times 1600 = 8^3\)	M1	oe eg \(h = \mathrm{k}m^{1/3}\)
(k =) \(1600 \div 8^3\) or 3.125 or (c =) \(8^3 \div 1600\) or 0.32	M1dep	oe eg \(\dfrac{1600}{512}\) or \(\dfrac{25}{8}\) \(\dfrac{512}{1600}\) or \(\dfrac{8}{25}\)
\(m = 3.125 \times h^3\) or \(0.32 \times m = h^3\)	A1	oe equation

Mark scheme (b)

Answer	Mark	Comments
their \(3.125 \times 12^3\) their \(3.125 \times 1728\) or \(1600 \times \left(\dfrac{12}{8}\right)^3\) or \(12^3 \div\) their 0.32 or \(1728 \div 0.32\) or \(1600 \div \left(\dfrac{8}{12}\right)^3\)	M1	oe
5400	A1ft	oe ft their 3.125 provided using \(m =\) their \(3.125 \times h^3\)

Must use \(\times\, 12^3\) or \(\times\) 1728 or \(\times \left(\dfrac{12}{8}\right)^3\) for M1
If in part (a) \(m = \mathrm{k} \times h\) \(1600 = \mathrm{k} \times 8\) \(m = 200h\) and in part (b) \(m = 200 \times 12\), \(m = 2400\)	M0 part (a) M0 part (b)
If in part (a) \(m = \mathrm{k} \times h\) \(1600 = \mathrm{k} \times 8\) \(m = 200h\) and in part (b) \(m = 200 \times 12^3\), \(m = 345\,600\)	M0 part (a) M1A1ft part (b)