(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?
Explanation
(a) By the use of Pythagoras theorem, the hypotenuse side is 13 and also sin x = \(\frac{5}{13}\) and cos x = \(\frac{12}{13}\) Substituting and simplifying yields \(\frac{65}{161}\)
(b) we have (11 + 2x)(14 + 2x) - (11 x 14) =186.
Expanding and simplifying to get the quadratic equation 2x\(^2\) + 25x - 93 = 0.
Solving the quadratic equation results in x = 3 m as the width of the path.