Question

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45°$ After walking a distance of $80$ meters towards the top, up a slope inclined at an angle of $30°$ to the horizontal plane, the angle of elevation of the top of the hill becomes $75°$. Then the height of the hill (in meters) is:

Answer 1

Find the height of the hill:

Using given data to draw a figure

By observing the figure,

Find $\tan \left(75°\right)$

Where $\tan \left(\theta \right)=\frac{\text{opposite side}}{\text{adjacent}}$

$$\begin{gathered} \tan {75}^{°}=\frac{h}{h+40-40\sqrt{3}} \\ \Rightarrow \frac{2+\sqrt{3}}{1}=\frac{h}{h+40-40\sqrt{3}} \\ \Rightarrow \left(h+40-40\sqrt{3}\right)\left(2+\sqrt{3}\right)=h \\ \Rightarrow 2h+80-80\sqrt{3}+h\sqrt{3}+40\sqrt{3}-120=h \\ \Rightarrow h(\sqrt{3}+1)=40+40\sqrt{3} \\ \Rightarrow h=40 \end{gathered}$$

Height of the hill

l$$\begin{gathered} =h+40 \\ =40+40 \\ =80 \end{gathered}$$

Hence, height of the hill is $80m$..