From the top of a building 20 m high, the angles of depression of the top and the bottom of a tower standing on the same horizontal ground are observed to be 30° and 60°, respectively. The height (in meter) of the tower is: (Round off your answer to two decimal places.)
13.33 m
Let the tower's height be h and the horizontal distance be d.
Depression to the tower's bottom is 60°: \(\tan60^\circ = \dfrac{20}{d} \Rightarrow d = \dfrac{20}{\sqrt3}\).
Depression to the tower's top is 30°: the vertical gap between the building top and tower top is \(20-h\), so \(\tan30^\circ = \dfrac{20-h}{d}\).
\(20-h = \dfrac{d}{\sqrt3} = \dfrac{20/\sqrt3}{\sqrt3} = \dfrac{20}{3}\).
\(h = 20-\dfrac{20}{3} = \dfrac{40}{3} \approx 13.33\).
Hence, the height of the tower is approximately 13.33 m.
Similar Questions
- The angle of elevation of the sun when the length of the shadow of a pole is equal to its height is:
- A tree broke at a height of 8 m from its foot and the broken upper part touches the ground at a point of 6 m from its foot. Find the height of the tree.
- An observer at the top of a tower observes that two cars are running towards the foot of the tower at a distance of 120 m from each other making angles of depression $\alpha$ and $\beta$ such that $\alpha > \beta$ and $\tan \alpha = \sqrt{3}$ and $\tan \beta = \frac{1}{\sqrt{3}}$. Find the height of the tower.
- The angle of elevation of a ladder leaning against a wall is $45^\circ$. The foot of the ladder is $4\sqrt{2}$ metres away from wall. The length of the ladder is:
- From a point Y on a level ground, the angle of elevation of the top of a lamp post is $45^\circ$. If the distance of point Y from the foot of the lamp post is 80 m, the height of the lamp post will be:
- The top of a tower makes complementary angles of elevation from two points at the distances of 25 m and 16 m from its foot on the ground. Find the height of the tower.
- A kite is flying with a thread of length 296 m, making an angle of elevation measuring 30° at a point of hand of a person of height 2m from the ground. Find the height of the kite from the ground.
- The angle of elevation of a lamp post changes from $30^{\circ}$ to $60^{\circ}$ when a person walks 30 m towards it. Find the height of the lamp post.
- The angles of depression of two houses of the same height from the top of a building are $45^{\circ}$ and $30^{\circ}$ towards the east. If the two houses are 50 m apart, what will be the height of the building in metres?
- A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches the base of a window 6 m above the ground. Find the length of the ladder.
Important Questions from Heights and Distances
If x is the distance of P from the bottom of the pillar, then consider the following statements :
1. x can take two values which are in the ratio 1 : 3
2. x can be equal to the height of the flagstaff
Which of the statements given above is/are correct?
What is a possible value of tan θ ?
A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are 30° and 45° respectively. Which one of the following gives the best approximation to the height of the tower?
Two poles are 10 m and 20 m high. The line joining their tops makes an angle of 15° with the horizontal. The distance between the poles is approximately equal to
The angle of elevation of the top of a tower from a point 20 m away from its base is 45 °. What is the height of the tower?
