(a) If tan x = \(\frac{5}{12}\), \(0^o\). < x < 90°, evaluate, without using Mathematical tables or calculator, \(\frac{sin x}{(sin x)^2 + cosx}\) (b) The d...
Question 1 Report
(a) If tan x = \(\frac{5}{12}\), \(0^o\). < x < 90°, evaluate, without using Mathematical tables or calculator, \(\frac{sin x}{(sin x)^2 + cosx}\)
(b) The diagram shows a rectangular lawn measuring 14m by 11m. A path of uniform width \(x\)m surrounds it. If the total area of the path is 186 m\(^2\), how wide is the path?
Answer Details
(a) \(\tan x = \frac{5}{12}\) gives opposite 5, adjacent 12, hypotenuse \(\sqrt{5^2 + 12^2} = 13\).
So \(\sin x = \frac{5}{13}\) and \(\cos x = \frac{12}{13}\).
\[\frac{\sin x}{\sin^2 x + \cos x} = \frac{\tfrac{5}{13}}{\tfrac{25}{169} + \tfrac{12}{13}} = \frac{\tfrac{5}{13}}{\tfrac{25 + 156}{169}} = \frac{5}{13} \times \frac{169}{181} = \frac{65}{181}\](b) Lawn is \(14 \times 11\); a path of width \(x\) all round gives an outer rectangle \((14 + 2x)\) by \((11 + 2x)\).
\[(14 + 2x)(11 + 2x) - 14 \times 11 = 186\] \[154 + 50x + 4x^2 - 154 = 186 \Rightarrow 4x^2 + 50x - 186 = 0\] \[2x^2 + 25x - 93 = 0\]Discriminant \(= 25^2 + 4(2)(93) = 625 + 744 = 1369 = 37^2\).
\[x = \frac{-25 + 37}{4} = 3\]The path is \(3\) m wide (the negative root is rejected).
Download The App On Google Playstore
Everything you need to excel in your exams
