Question

At the foot of a mountain the elevation of its summit is $45°$. After ascending $1000m$ towards the mountain up a slope of $30°$ inclination, the elevation is found to be $60°$. Find the height of the mountain.

Answer 1

Step 1. Draw a diagram for the given problem

Let $AB$$=h$be the height of the mountain.

$\angle AOB$$=$$45°$ is the angle of elevation of the summit.

$OC=$ $1000m$, $\angle COE=$ $30°$ and $\angle BCD=$$60°$.

Step 2. Find the length of the line $CE$

Consider the triangle $OCE$

$\sin \left(\theta \right)=\frac{Oppositesideto\theta }{hypotenuse}$

In triangle $\mathrm{OCE}$

$\sin 30°=\frac{CE}{OC}$

$\Rightarrow \frac{1}{2}=\frac{CE}{1000}$ [as $\sin 30°=\frac{1}{2}$ and $OC=1000$]

$\Rightarrow \frac{1000}{2}=CE$ [by cross multiplication]

$\Rightarrow CE=500m$

Step 3. Find the length of the line $OE$

Cosine function is the ratio of base to hypotenuse.

In triangle $OCE$

$$\begin{gathered} \cos \theta =\frac{adjacent}{hypotenuse} \\ \Rightarrow \cos 30°=\frac{OE}{OC} \end{gathered}$$

$\Rightarrow \frac{\sqrt{3}}{2}=\frac{OE}{1000}$ [as $\sin 30°=\frac{\sqrt{3}}{2}$ and $OC=1000m$]

$\Rightarrow \frac{1000\sqrt{3}}{2}=OE$ [by cross multiplication]

$\Rightarrow OE=500\sqrt{3}m$

Step 4. Find the relation between line $OA$ and line $AB$

In triangle $AOB$

$$\begin{gathered} \tan \theta =\frac{opposite}{adjacent} \\ \tan 45°=\frac{AB}{OA} \end{gathered}$$

$1=\frac{AB}{OA}$ [as $\tan 45°=1$]

$OA=AB=h$ [by cross multiplication]

Using this we get $CD=EA=OA-OE=h-500\sqrt{3}$ [as $AECD$ is a rectangle]

Now from the diagram, we get $BD=AB-AD=AB-CE=h-500$

Step 5. Find the height of the mountain

In triangle $BCD$,

$$\begin{gathered} \tan 60°=\frac{\mathrm{BD}}{\mathrm{CD}} \\ \Rightarrow \sqrt{3}=\frac{h-500}{h-500\sqrt{3}} \\ \Rightarrow \sqrt{3}h-1500=h-500 \\ \Rightarrow \mathrm{h}(\sqrt{3}-1)=1000 \end{gathered}$$\

$$\begin{gathered} \Rightarrow h=\frac{1000}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1} \\ \Rightarrow h=\frac{1000(\sqrt{3}+1)}{3-1} \\ \Rightarrow \mathrm{h}=500(\sqrt{3}+1) \\ \Rightarrow h=500\times 2.732=1366\mathrm{m} \\ \Rightarrow \mathrm{h}=1.366\mathrm{km} \end{gathered}$$

Hence, the height of the mountain is $1366\mathrm{m}$ or $1.366\mathrm{km}$.