Angles of elevation

From points A and B on level ground, the angles of elevation of the top of a building are 25° and 37°, respectively. If |AB| = 57m, calculate, to the nearest meter, the distances of the top of the building from A and B if they are both on the same side of the building.

Correct answer:

h =  69.73 m
d₁ =  165 m
d₂ =  116 m

Step-by-step explanation:

$$\begin{gathered} a=57\text{m} \\ A=25°\to \text{rad}=25°\cdot \frac{\pi }{180}=25°\cdot \frac{3.1415926}{180}=0.43633=5\pi /36 \\ B=37°\to \text{rad}=37°\cdot \frac{\pi }{180}=37°\cdot \frac{3.1415926}{180}=0.64577 \\ \tan A=\frac{h}{a+b} \\ \tan B=\frac{h}{b} \\ b=\frac{h}{\tan B} \\ a\tan A+b\tan A=h \\ a\tan A+\frac{h}{\tan B}\tan A=h \\ h=\frac{a\cdot \tan (A)}{1-\tan (A)/\tan (B)}=\frac{57\cdot \tan 0.4363}{1-\tan 0.4363/\tan 0.6458}=69.73\text{m} \end{gathered}$$

$$\begin{gathered} b=\frac{h}{\tan (B)}=\frac{69.728}{\tan 0.6458}\doteq 92.5322\text{m} \\ {d}_1=\sqrt{h^2+(a+b{)}^2}=\sqrt{69.728^2+(57+92.5322{)}^2}=165\text{m} \end{gathered}$$

$d_2=\sqrt{h^2+b^2}=\sqrt{69.728^2+92.5322^2}=116\text{m}$

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