Question

A tower 125 ft high is on the cliff on the bank of a river. From the top of the tower the angle of depression of a point on the opposite shore is 28°41' and from the base of the tower the angle of depression of the same point is 18°20' (a)Find the width of the river and (b) height of the cliff

Answer 1

Answer on question #35312 – Math – Geometry

A tower 125 ft high is on the cliff on the bank of a river. From the top of the tower the angle of depression of a point on the opposite shore is $28{}^{\circ}41'$ and from the base of the tower the angle of depression of the same point is $18{}^{\circ}20'$ (a) Find the width of the river and (b) height of the cliff

Answer

Let $x$ is the high of cliff. And $y$ is the width of river. Then we get the system of two equations.

$$\left\{ \begin{array}{c} \tan 28{}^{\circ}41 = \frac{x + 125}{y} \\ \tan 18{}^{\circ}20' = \frac{x}{y} \end{array} \right.$$ $$\left\{ \begin{array}{c} y \tan 28{}^{\circ}41 = x + 125 \\ x = y \tan 18{}^{\circ}20' \end{array} \right.$$ $$y \tan 28{}^{\circ}41 = y \tan 18{}^{\circ}20' + 125$$ $$y (\tan 28{}^{\circ}41 - \tan 18{}^{\circ}20') = 125$$ $$y = \frac{125}{(\tan 28{}^{\circ}41 - \tan 18{}^{\circ}20')} \approx 580,32 \text{ ft}.$$ $$x = 580,32 \tan 18{}^{\circ}20' \approx 192.26 \text{ ft}.$$

Answer: (a) 580,32 ft.; (b) 192.26 ft.