Depth angle

Determine the height of the cloud above the lake's surface if we see it from place A at an elevation angle of 20° 57'. From the same place A, we see its image in the lake at a depth angle of 24° 12'. Observation point A is 115 m above the lake level.

Correct answer:

h =  1438.0998 m

Step-by-step explanation:

$$\begin{gathered} \alpha =20+57/60=\frac{419}{20}=20.95{}^{\circ } \\ \beta =24+12/60=\frac{121}{5}=24.2{}^{\circ } \\ y=115\text{m} \\ \tan \alpha =(h-y):x \\ \tan \beta =(h+y):x \\ \frac{\tan \alpha }{\tan \beta }=\frac{h-y}{h+y} \\ (h+y)\cdot \tan \alpha =(h-y)\cdot \tan \beta \\ {t}_1=\tan \alpha =\tan 20.95°=0.382863=0.38286 \\ {t}_2=\tan \beta =\tan 24.2°=0.449418=0.44942 \\ h\cdot t_1+y\cdot t_1=h\cdot t_2-y\cdot t_2 \\ h(t_1-t_2)=-y\cdot t_1-y\cdot t_2 \\ h=y\cdot \frac{t_1+t_2}{t_2-t_1}=115\cdot \frac{0.3829+0.4494}{0.4494-0.3829}=1438.0998\text{m} \end{gathered}$$

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