When one end of a ladder, LM, is placed against a vertical wall at a point 5 metres above the ground, the ladder makes an angle of 37° with the horizontal ground.
(a) Represent this information in a diagram ;
(b) Calculate, correct to 3 significant figures, the length of the ladder ;
(c) If the foot of the ladder is pushed towards the wall by 2 metres, calculate,correct to the nearest degree, the angle which the ladder nows makes with the ground.
Explanation
(a) see diagram above
(b) From the above diagram,
\(\sin 37° = \frac{5}{|LM|}\)
\(|LM| = \frac{5}{\sin 37}\)
\(|LM| = \frac{5}{0.6018}\)
= \(8.308 m\)
\(\approxeq 8.31 m\)
\(\cos 37 = \frac{|AM|}{|LM|}\)
\(|AM| = 8.308 \times \cos 37\)
\(|AM| = 8.308 \times 0.7986\)
\(|AM| = 6.6348 m\)
(c) The new |AM| = 6.6348 m - 2 = 4.6348 m
\(\cos y = \frac{4.6348}{8.308}\)
\(\cos y = 0.5579\)
\(y = \cos^{-1} (0.5579)\)
\(y = 56.089°\)
\(y = 56°\) (to the nearest degree).