(a) Solve, for \(x\) and \(y\), the simultaneous equations \[3\log_2 x = y \qquad\text{and}\qquad \log_2 4x = y + 4.\] (b) Using the value of \(y\) obtained...
Question 1 Report
(a) Solve, for \(x\) and \(y\), the simultaneous equations
\[3\log_2 x = y \qquad\text{and}\qquad \log_2 4x = y + 4.\](b) Using the value of \(y\) obtained in part (a), express \(y^{2} - 10y + 25\) in the form \(2^{n}\), and hence find the value of \(n\).
Answer Details
This is a concurrent-forces (Lami's theorem) problem in which a load of 120 N is held in equilibrium by a tension \(T\) and a reaction \(R\), with the geometry giving angles of \(90^\circ\) and \(135^\circ\) opposite \(T\) and the load respectively.
(a) By Lami's theorem, \(\dfrac{T}{\sin 90^\circ}=\dfrac{120}{\sin 135^\circ}\), so
\[T=\frac{120\sin 90^\circ}{\sin 135^\circ}=\frac{120(1)}{0.7071}=169.71\ \text{N}.\](b) Similarly \(\dfrac{R}{\sin 135^\circ}=\dfrac{120}{\sin 135^\circ}\), giving
\[R=120\times\frac{\sin 135^\circ}{\sin 135^\circ}=120\ \text{N}.\]Hence \(T\approx\textbf{169.71 N}\) and \(R=\textbf{120 N}\), confirming the stated working. (The full configuration/diagram was not supplied, so the angle assignments are taken from the given ratios.)
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