### Variation; similar figures

Question Sample Titled 'Variation; similar figures'

Let ${H}$ $\text{seconds}$ be the required time of coating metal onto a suitcase of surface area ${X}$ $\text{m}^{{2}}$ . It is given that ${H}$ is the sum of two parts, one part is a constant and the other part varies as ${X}$ . When ${X}={3}$ , ${H}={1960}$ ; when ${X}={11}$ , ${H}={5160}$ .

 (a) Find the required time of coating metal onto a suitcase of surface area of ${15}$ $\text{m}^{{2}}$ . (4 marks) (b) There is a larger suitcase which is similar to the suitcase described in (a). If the volume of the larger suitcase is ${8}$ times that of the suitcase described in (a), find the required time of coating metal onto the larger suitcase. (2 marks)

 (a) Let ${H}={y}+{p}{X}$ , where ${y}$ and ${p}$ are non-zero constants. 1A So, we have ${y}+{3}{p}={1960}$ and ${y}+{11}{p}={5160}$ . 1M Solving, we have ${y}={760}$ and ${p}={400}$ . 1A for both correct The required cost$={760}+{400}{\left({15}\right)}$ $={6760}$ $\text{seconds}$ 1A u-1 for missing unit  (b) By the concept of similar figures, the surface area of the larger suitcase is ${8}^{{\tfrac{{2}}{{3}}}}$ times that of the suitcase decribed in (a) since the volume of the larger suitcase is ${8}$ times that of the suitcase described in (a). 1M The required cost $={760}+{400}{\left({15}\right)}\cdot{8}^{{\tfrac{{2}}{{3}}}}$  $={24760}$ $\text{seconds}$ 1A u-1 for missing unit

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