$A$ tree is broken by wind,its upper part touches the ground at a point $10 \ m$ from the foot of the tree and makes an angle of $45^\circ$ with the ground. The total length of the tree is:

$A$ vertical pole (more than $100 \, ft$ high) consists of two portions,the lower being one-third of the whole. If the upper portion subtends an angle $\tan^{-1}(\frac{1}{2})$ at a point in a horizontal plane through the foot of the pole and at a distance of $40 \, ft$ from it,then the height of the pole is......$ft$.

In the figure,$\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $\triangle CAB$ is $2\sqrt{3}-3 \text{ unit}^2$,when $\frac{\theta_2}{\theta_1}$ is the largest,then the perimeter (in unit) of $\triangle CED$ is equal to $...........$.

$A$ spherical gas balloon of radius $16 \ m$ subtends an angle $60^{\circ}$ at the eye of the observer $A$,while the angle of elevation of its center from the eye of $A$ is $75^{\circ}$. Then the height (in $m$) of the top most point of the balloon from the level of the observer's eye is:

The shadow of a tower is found to be $60 \ m$ shorter when the sun's altitude changes from $30^{\circ}$ to $60^{\circ}$. The height of the tower from the ground is approximately equal to......$m$

$A$ man from the top of a $100 \ m$ high tower sees a car moving towards the tower at an angle of depression of $30^\circ$. After some time,the angle of depression becomes $60^\circ$. The distance (in meters) travelled by the car during the time is