The horizontal distance between two trees of different heights is $60 \mathrm{m}$. The angle of depression of the top of the first tree, when seen from the top of the second tree is $45°$. If the height of the second tree is $80 \mathrm{m}$, find the height of the first tree.

$\mathrm{PQ}$ is a post of given height $\mathrm{a}$ and $\mathrm{AB}$ is a tower at some distance, $\mathrm{\alpha }$ and $\mathrm{\beta }$ are the angles of elevation of $\mathrm{B}$, the top of the tower, at $\mathrm{P}$ and $\mathrm{Q}$ respectively. Find the height of the tower and its distance from the post.

A carpenter makes stools for electrician with a square top of side $0.5 \mathrm{m}$ and at a height $1.5 \mathrm{m}$ above the ground. Also each leg is inclined at an angle of $60°$ to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances in metres.