(a) Simplify : \((2a + b)^{2} - (b - 2a)^{2}\)
(b) Given that \(S = K\sqrt{m^{2} + n^{2}}\); (i) make m the subject of the relations ; (ii) if S = 12.2, K = 0.02 and n = 1.1, find, correct to the nearest whole number, the positive value of m.
(a) \((2a + b)^{2} - (b - 2a)^{2}\)
Using the method of difference of two squares,
\((2a + b)^{2} - (b - 2a)^{2} = ((2a + b) + (b - 2a))((2a + b) - (b - 2a))\)
= \((2b)(4a)\)
= \(8ab\).
(b)(i) \(S = K\sqrt{m^{2} + n^{2}}\)
\(S^{2} = K^{2}[m^{2} + n^{2}]\)
\(\frac{S^{2}}{K^{2}} = m^{2} + n^{2}\)
\(m^{2} = \frac{S^{2}}{K^{2}} - n^{2}\)
\(m = \sqrt{\frac{S^{2}}{K^{2}} - n^{2}}\)
(ii) When S = 12.2, K = 0.02, n = 1.1
\(m = \sqrt{\frac{12.2^{2}}{0.02^{2}} - (1.1^{2})}\)
\(m = \sqrt{\frac{148.84}{0.0004} - 1.21}\)
\(m = \sqrt{372,098.79}\)
= \(609.999 \approxeq 610\)
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