(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.