The height of a tower is $30 \ m$. Find the length of the shadow of the tower,when the angle of elevation of the sun is $45^\circ$. (in $m$)

$A$ cable tied to the top of an electric pole is fixed at a point on the ground. The length of the cable is $22\, m$ and it makes an angle of measure $30^{\circ}$ with the ground. Find the height of the electric pole (in $m$).

From the top of a building $h \ m$ high,the angle of elevation of the top of a pole is found to be $\alpha$,while the angle of depression of the base of the pole is found to be $\beta$. Prove that the height of the pole is $h(1 + \tan \alpha \cdot \cot \beta) \ m$.

From a point on the ground,the angle of elevation of the top of a tower is found to be $30^{\circ}$. Walking $30\, m$ towards the tower from that point,the angle of elevation of the top of the tower is found to be $60^{\circ}$. Find the height of the tower (in $m$).

The lower window of a house is at a height of $2\, m$ above the ground and its upper window is $4\, m$ vertically above the lower window. At a certain instant,the angles of elevation of a balloon from these windows are observed to be $60^{\circ}$ and $30^{\circ}$,respectively. Find the height of the balloon above the ground (in $m$).

Watching from the top of a building $40 \, m$ high,the angle of depression of a moving car increases from $30^{\circ}$ to $45^{\circ}$ in some interval of time. Find the distance covered by the car in that interval of time. (in $m$)