Some Applicatio...

From the top of a tree a man observes the angle of depression of a moving car is $$30^{o}$$ and after $$3$$ minutes he finds the angle of depression is $$60^{o}$$. How much time will the car take to reach the tree?

Assertion ($$A$$): ladder rests against a wall at an angle $$30^{0}$$ to the horizontal. Its foot is pulled away through a distance $$x$$' so that it slides a distance $$y$$' down the wall finally making an angle $$60^{0}$$ with the horizontal then $$x=y$$.
Reason ($$R$$): $$A$$ ladder rests against a wall at angle $$\alpha $$ to the horizontal. Its foot is pulled a way through a distence $$a$$' so that it slides a distence $$b$$' down the wall, finally making an angle $$\beta$$ with the horizonal then $$\displaystyle \tan(\frac{\alpha+\beta}{2})=b/a$$

A flag staff of $$5$$ mts high stands on a building of $$25 $$ mt high. At an observer at a height of $$30$$ mt the flag staff and the building subtend equal angles. The distance of the observer from the top of the flag staff is

Flag-staff of length $$d$$ stands on a tower of height $$h$$. lf at a point on the ground the angles of elevation of the tower and the top of the flag-staff be $$\alpha,\ \beta$$ respectively, then $$h=$$

A ladder $$20\ m$$ long reaches a point $$20\ m$$ below the top of a flag and makes an angle $$60^\circ$$ with the horizontal. Find the length of the flagstaff, in meters

The angle of elevation of a cloud from a point $$h$$ metres above a lake is $$\Theta$$. The angle of depression of its reflection in lake is $$45^{ }$$ The height of the cloud is

$$PQ$$ is a vertical tower. $$A,\ B,\ C$$ are three points in a horizontal line through $$Q$$, the foot of the tower. If the angles of elevation of the top of the tower from $$A,\ B,\ C$$ are $$\alpha,\ \beta,\ \gamma$$ respectively, then $$BC\cot\alpha-CA\cot\beta+AB \cot$$ $$\gamma=$$

A man on a cliff observes a boat at an angle of depression $$30^{o}$$ which is sailing towards the shore to the point immediately beneath him. $$3$$ minutes later the angle of depression of the boat is found to be $$60^{o}$$. Assuming that the boat sails at a uniform speed, the time taken by the boat to reach the shore is:

$$A$$ tower standing at point $$A$$ leans towards west making an angle $$\alpha$$ with the vertical. The angular elevation of $$B$$, the top most point of the tower is $$\beta$$ as observed from a point $$C$$ due east of $$A$$ at a distance $$d$$ from $$A$$. lf the angular elevation of $$B$$ from a point due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma$$, then $$ 2\tan\alpha$$ can be written as

Assertion (A): The shadow of a tower on a level plane is found to be 60 meters longer when sun's altitude is $$30^{0}$$ than that when it is $$45^{0}$$ Then the height of the tower is $$30$$ $$(\sqrt{3}+1)$$ {meters}