(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.
Explanation
(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 \(m = \sqrt{\frac{12.2^{2}}{0.02^{2}} - (1.1^{2})}\) \(m = \sqrt{\frac{148.84}{0.0004} - 1.21}\) \(m = \sqrt{372,098.79}\) = \(609.999 \approxeq 610\)
