(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 =...

Answer Details

(a) Simplify \((2a + b)^2 - (b - 2a)^2\). Note that \((b - 2a)^2 = (2a - b)^2\). This is a difference of two squares, \(P^2 - R^2 = (P + R)(P - R)\), with \(P = 2a + b\) and \(R = 2a - b\):

\[(2a + b)^2 - (2a - b)^2 = \big[(2a + b) + (2a - b)\big]\big[(2a + b) - (2a - b)\big].\]

\[= (4a)(2b) = 8ab.\]

(b) Given \(S = K\sqrt{m^2 + n^2}\).

(i) Make \(m\) the subject.

\[\frac{S}{K} = \sqrt{m^2 + n^2} \Rightarrow \left(\frac{S}{K}\right)^2 = m^2 + n^2.\]

\[m^2 = \left(\frac{S}{K}\right)^2 - n^2 \Rightarrow m = \sqrt{\left(\frac{S}{K}\right)^2 - n^2}.\]

(ii) With \(S = 12.2, K = 0.02, n = 1.1\):

\[\frac{S}{K} = \frac{12.2}{0.02} = 610, \qquad \left(\frac{S}{K}\right)^2 = 372100, \qquad n^2 = 1.21.\]

\[m = \sqrt{372100 - 1.21} = \sqrt{372098.79} = 609.999 \approx 610.\]

To the nearest whole number, the positive value of \(m = \mathbf{610}\).