2022HSCShort answer

Q16 (c) (i)

2 marks

Question

It is given that for positive numbers $x_1,x_2,x_3,\ldots,x_n$ with arithmetic mean $A$ ,
$\frac{x_1x_2x_3\cdots x_n}{A^n}\le1.$
(Do NOT prove this.)

Suppose a rectangular prism has dimensions $a,b,c$ and surface area $S$ . Show that
$abc\le\left(\frac S6\right)^{3/2}.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Attempts to apply given information with three numbers, or equivalent merit	1

Sample answer:
Consider $ab$ , $ac$ , $bc$ .
$S=2(ab+ac+bc)$
From the given result,
$ab\times ac\times bc\le\left(\frac{ab+ac+bc}{3}\right)^3$
$(abc)^2\le\left(\frac S6\right)^3$
$abc\le\left(\frac S6\right)^{3/2}$

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Syllabus alignment

Proof

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.

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