Angle of Depression

A man on the deck of a ship is $16 \mathrm{m}$ above water level. He observes that the angle of elevation of the top of a cliff is $45°$ and the angle of depression of the base is $30°$. Calculate the distance of the cliff from the ship and the height of the cliff. 

From a building $60$ meters high, the angles of depression of the top and bottom of a lamp post are $30°$ and $60°$ respectively. Find the distance between the lamp post and the building. Also, find the difference of height between the building and the lamp post. 

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be $\mathrm{\alpha }$ and $\mathrm{\beta }$. Show that the height of the aeroplane above the road is $\frac{\mathrm{tan\alpha tan\beta }}{\mathrm{tan\alpha }+\mathrm{tan\beta }}$.

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.

A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes $12$ minutes for the angle of depression to change from $30°$ to $45°$ how soon after this, will the car reach the tower?

A parachutist is descending vertically and makes angles of depression of $45°$ and $60°$ at two observation points $100 \mathrm{m}$ apart from each other on the left side of himself. Find, in metres, the approximate height from which he falls and also find, in metres the approximate distance of the point, where he falls on the ground from the first observation point. 

From the terrace of a house $8 \mathrm{m}$ high, the angle of elevation of the top of a tower is $45°$ and the angle of depression of its reflection in a lake is found to be $60°.$ Determine the height of the tower from the ground. 

From a point $100 \mathrm{m}$ above the surface of a lake the angular elevation of the peak of a mountain is found to be $30°$ and the angle of depression of the image of the peak is $45°.$ Find the height of the peak. 

The pilot of an aircraft flying horizontally at a speed of $1200\mathrm{km}/\mathrm{hr}$ observes that the angles of depression of a point on the ground changes from $30°$ to $45°$ in $15$ seconds. If the height at which the aircraft is flying is $h \mathrm{km}$, then find the value of $h$, correct to two decimal places. [Take $\sqrt{3}=1.732$]

The horizontal distance between two towers is $70 \mathrm{m}$. The angles of depression of the top of the first tower, when seen from the top of the second tower is $30°$. The height of the second tower is $120 \mathrm{m}$. If the height of the first tower is $k\mathit{ }\mathrm{m}$, then find the value of $k$ (rounded off to one decimal place) considering $\sqrt{3}=1.732$. 

A man on the roof of a house, which is $10 \mathrm{m}$ high, observes the angle of elevation of the top of a building as $45°$ and angle of depression of the base of the building as $30°$. Find the height of the building and its distance from the house.

The angles of depression of the top and the bottom of a building $50$ meters high as observed from the top of a tower are $30°$ and $60°$ respectively. Find the height of the tower and also the horizontal distance between the building and the tower.

From the top of a building $12 \mathrm{m}$ high, the angle of elevation of the top of a tower is found to be $45°$ and the angle of depression of the base of the tower as $30°.$ Find the height of the tower and its distance on the ground from the building.

The angle of depression of two ships from the top of a lighthouse are $45°$ and $30°$ towards east. If the ships are $200$ meters apart, find the height of the light house.

A player sitting on the top of a tower of height $20 \mathrm{m}$ observes the angle of depression of a ball lying on the ground as $60°.$ If the distance between the foot of the tower and the ball is $k\mathit{ }\mathrm{m}$, then find the value of $k$ (rounded off to two decimal places) considering $\sqrt{3}=1.732$.

From the top of a $10 \mathrm{m}$ tall tower the angle of depression of a point on a ground was found to be $60°$. The point from the base of the tower is $\frac{k\sqrt{3}}{3} \mathrm{m}$. Find $k$.

The angle of depression from the top of a tower $12 \mathrm{m}$ high, at a point on the ground is $30°$. The distance of the point from the top of the tower is