Sample QuestionsHeights and Distance questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the angle of depression of the top and the bottom of a tower as observed from the top of a $h$ metre high cliff are $30^{\circ}$ and $60^{\circ}$ respectively, prove that the height of the tower is $\frac{2 h}{3}$.
View full solution →The angles of elevation of the top of a vertical tower from two points, at a distance a and b (a > b) from the base and in the same straight line with it are complementary. Find the height of the tower
A man standing on the top of a vertical tower observes a car moving towards the tower at a uniform speed. If it takes 10 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the tower?
A tree is broken by the wind. Find the total height of the tree if the top struck the ground at an angle of 30° and at a distance of 18 m from the foot of the pole.
A circus artist is climbing a 30 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the distance of the pole to the peg in the ground, if the angle made by the rope with the ground level is 30°.
There is a building of height 7 m next to a cable tower of unknown height. From the top of the building, the angle of elevation of the top of the tower is $60^{\circ}$ and the angle of depression to the foot of the tower is $45^{\circ}$. Find the height of the cable tower.
View full solution →An aeroplane is flying at a height of 300 m above the ground. Flying at this height, the angles of depression from the plane to two points on both banks of a river in opposite directions are $45^{\circ}$ and $60^{\circ}$ respectively. Find the width of the river (Use $\sqrt{3}=1.73$ )
View full solution →Two poles of equal heights are standing opposite to each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of poles are $60^{\circ}$ and $30^{\circ}$ respectively. Find the height of poles and the distances of the point from the poles.
View full solution →Two towers AB and CD are standing at some distance apart. From the top of tower AB , the angle of depression of the foot of tower CD is $30^{\circ}$. From the top of tower CD , the angle of depression of the foot of tower AB is $60^{\circ}$. If the height of tower CD is ' $h$ ' m , then prove that the height of tower AB is $\frac{h}{3} \mathrm{~m}$.
View full solution →An observer measures angles of elevation of two towers of equal height from a point between the towers. The angles of elevation of the tops of the two towers from this point are $60^{\circ}$ and $30^{\circ}$. If this point is at a distance of 120 m from the first tower, find the distance between the towers.
View full solution →The angle of elevation of the top of a tower from a certain point is 30°. If the observer moves 20 m towards the tower, the angle of elevation of the top increases by 20°. Find the height of the tower.
The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 40° and the angle of depression of its shadow in water of the lake is 65°. Find the height of the cloud from the surface of the water.
From the top of a tower 100 m high, a man observes two cars on opposite sides of the tower and in the same straight line with its base, with angles of depression 35° and 50°. Find the distance between the two cars.
A man observes the angle of elevation of the top of a building to be 30°. He walks towards it in horizontal line through its base. On covering 60 m, the angle of elevation changed to 60°. Find the height of the building.
The angle of elevation of the top of a vertical tower PQ from a point X on the ground is $60^{\circ}$. From a point $\mathrm{Y}, 40 \mathrm{~m}$ vertically above X , the angle of elevation of the top Q of the tower is $45^{\circ}$. Find the height of the tower PQ and distance PX. (Use $\sqrt{3}=1.73$ ).
View full solution →The angles of depression of two ships from the top of a lighthouse are $45^{\circ}$ and $30^{\circ}$ towards east. If the ships are 100 m apart, the height of the lighthouse is:
- A
$\frac{20}{(\sqrt{3}+1)} \mathrm{m}$
- B
$\frac{20}{(\sqrt{3}-1)} \mathrm{m}$
- C
$50(\sqrt{3}-1) \mathrm{m}$
- D
$50(\sqrt{3}+1) \mathrm{m}$
View full solution →From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $30^{\circ}$ and $45^{\circ}$ respectively. If bridge is at the height of 30 m from the bank, then the width of the river is:
- A
- B
$30 \sqrt{3} \mathrm{~m}$
- C
$30(\sqrt{3}+1) \mathrm{m}$
- D
$25(\sqrt{3}-1) \mathrm{m}$
View full solution →The shadow of a tower, standing on a level ground, is found to be 40 m longer when Sun's altitude is $30^{\circ}$ than when it was $60^{\circ}$. Then height of the tower is:
View full solution →An observer 1.5 metres tall is 18.5 metres away from the tower. If the angle of elevation of the top of the tower from his eye is $45^{\circ}$, the height of the tower is:
View full solution →A bridge, in the shape of a straight path, across a river, makes an angle of $60^{\circ}$ with the width of the river. If length of the bridge is 100 metres, then the width of the river is:
View full solution →Assertion (A) : At some time of a day, if the length of the shadow of a tower is equal to its height, then the sun's elevation is $45^{\circ}$.
Reason (R) : The angle which the line of sight makes with the horizontal line passing through the eye of the observer, when the object is above the observer, is called the angle of elevation.
- A
- ✓
- C
Both A and R are true, and R is the correct reason for A .
- D
Both A and R are true, and R is incorrect reason for A .
Answer: B.
View full solution →Assertion (A) : When we move towards the object, angle of elevation decreases.
Reason (R) : As we move towards the object, it subtends large angle at our eyes than before.
- A
- B
- C
Both A and R are true, and R is the correct reason for A .
- ✓
Both A and R are true, and R is incorrect reason for A .
Answer: D.
View full solution →