A tree is cut partially and made to fall on ground. The tree however does not fall completely and is still attached to its cut part. The tree top touches the ground at a point 10m from foot of the tree making an angle of 30°. What is the length of the tree?

A tree is cut partially and made to fall on ground. The tree however does not fall completely and is still attached to its cut part. The tree top touches the ground at a point 10m from foot of the tree making an angle of 30°. What is the length of the tree? Correct Answer $$10\sqrt 3 \,{\text{m}}$$

$$\eqalign{ & {\text{in}}\,\Delta MNQ,\tan {30^ \circ } = \frac{{MQ}}{{NQ}} \cr & \therefore \frac{1}{{\sqrt 3 }} = \frac{{MQ}}{{10}} \cr & \therefore MQ = \frac{{10}}{{\sqrt 3 }} \cr & {\text{Also}}\,{\text{by}}\,{\text{Pythagoras}}\,{\text{theorem}} \cr & M{N^2} = M{Q^2} + N{Q^2} \cr & \therefore {L^2} = \frac{{100}}{3} + 100 \cr & \therefore L = \frac{{20}}{{\sqrt 3 }} \cr & \therefore {\text{Height}}\,{\text{of}}\,{\text{tree}} \cr & = L + MQ \cr & = \frac{{20}}{{\sqrt 3 }} + \frac{{10}}{{\sqrt 3 }} \cr & = \frac{{30}}{{\sqrt 3 }} \cr & = \frac{{3 \times 10}}{{\sqrt 3 }} \cr & = 10\sqrt 3 \,{\text{m}} \cr} $$