# Question #04430

Dec 11, 2017

The viewing angle between the base and top of the castle wall from the person on the boat will be

$\theta = \beta - \alpha = {19.8}^{\circ} - {9.1}^{\circ} = {10.7}^{\circ}$

#### Explanation:

Let the angle of elevation of the top of the cliff or base of the castle from the boat be $\alpha$ and the angle of elevation of the top of the castle from the boat be $\beta$

Again the perpendicular distance of the boat from the castle or cliff is $200 m$

Height of the top of castle wall is $32 m + 40 m = 72 m$

Height of the base of castle wall or top of the cliff is $32 m$

So $\tan \beta = \frac{72}{200} = 0.36$

$\implies \beta = {\tan}^{-} 1 \left(0.36\right) = {19.8}^{\circ}$

And $\tan \alpha = \frac{32}{200} = 0.16$

$\implies \alpha = {\tan}^{-} 1 \left(0.16\right) = {9.1}^{\circ}$

Hence the viewing angle between the base and top of the castle wall from the person on the boat will be

$\theta = \beta - \alpha = {19.8}^{\circ} - {9.1}^{\circ} = {10.7}^{\circ}$

Dec 11, 2017

${10}^{\circ} 42 ' 31 ' '$ or ${10.7}^{\circ}$

#### Explanation:

Viewing $\angle$ between bottom of wall and level line to
the foot of the cliff:-$= \frac{32}{200} = \tan \theta = {9}^{\circ} 5 ' 25 ' '$

Vertical height between the foot of the cliff and top of wall

$= 40 + 32 = 72 m$

Viewing $\angle$ between bottom of cliffl and level line to
the top of the wall:-$= \frac{72}{200} = \tan \theta = {19}^{\circ} 47 ' 56 ' '$

$=$the viewing angle between the base and the top of
the castle wall :-

$\left({19}^{\circ} 47 ' 56 ' '\right) - \left({9}^{\circ} 5 ' 25 ' '\right) = {10}^{\circ} 42 ' 31 ' '$ or ${10.7}^{\circ}$