1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 45° and 30° respectively. Find the height of the hill. [CBSE 2017]

Open in App
Solution

## Let PQ be the hill of height h km. Let R and S be two consecutive kilometre stones, so the distance between them is 1 km. Let QR = x km. $\mathrm{In}∆\mathrm{PQR},\phantom{\rule{0ex}{0ex}}\mathrm{tan}45°=\frac{\mathrm{PQ}}{\mathrm{QR}}\phantom{\rule{0ex}{0ex}}⇒1=\frac{h}{x}\phantom{\rule{0ex}{0ex}}⇒h=x...\left(i\right)$ $\mathrm{In}∆\mathrm{PQS},\phantom{\rule{0ex}{0ex}}\mathrm{tan}30°=\frac{\mathrm{PQ}}{\mathrm{QS}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{\sqrt{3}}=\frac{h}{x+1}\phantom{\rule{0ex}{0ex}}⇒\sqrt{3}h=x+1...\left(\mathrm{ii}\right)$ From equation (i) and (ii) we get, $\sqrt{3}h=h+1\phantom{\rule{0ex}{0ex}}⇒h\left(\sqrt{3}-1\right)=1\phantom{\rule{0ex}{0ex}}⇒h=\frac{1}{\sqrt{3}-1}=\frac{\sqrt{3}+1}{\left(\sqrt{3}-1\right)\left(\sqrt{3}+1\right)}\phantom{\rule{0ex}{0ex}}⇒h=\frac{\sqrt{3}+1}{2}=\frac{2.73}{2}=1.365\mathrm{km}$ Hence, the height of the hill is 1.365 km.

Suggest Corrections
5
Join BYJU'S Learning Program
Related Videos
Pythogoras Theorem
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program