Angle of Elevation Problem: Lamp Post Height
This solution details finding the lamp post's height using trigonometry based on the angle of elevation and distance.
Lamp Post Scenario Analysis
We have a right-angled triangle formed by:
- The lamp post (height = '$h$', opposite side).
- The distance from point Y to the lamp post's foot (80 m, adjacent side).
- The line of sight from Y to the top (hypotenuse).
Given:
- Angle of elevation ($\theta$) = $45^\circ$.
- Adjacent side = 80 m.
- Find: Opposite side ($h$).
Height Calculation using Tangent
The tangent function connects the angle to the opposite and adjacent sides:
$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $
Substitute the known values:
$ \tan(45^\circ) = \frac{h}{80 \text{ m}} $
Since $\tan(45^\circ) = 1$:
$ 1 = \frac{h}{80 \text{ m}} $
Solving for '$h$':
$ h = 1 \times 80 \text{ m} $
$ h = 80 \text{ m} $
Result: Lamp Post Height
The height of the lamp post is 80 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:
- 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.
- If the height of a pole is $6\sqrt{3}$ metre and the length of its shadow is 6 metre, then the angle of elevation of the sun is:
Important Questions from Heights and Distances
Two ships are sailing in the sea on the two sides of a lighthouse. The angles of elevation of the top of the lighthouse as observed from the ships are 45 ° and 60° respectively. If the lighthouse is 81 m high, then the distance between two ships is:
The horizontal distance between two towers is 40√3 m. The angle of depression of the top of the first tower when seen from the top of the second tower is 30°. If the height of the second tower is 130 m, find the height of the first tower.
The angle of elevation of a ladder leaning against a house is 60° and the foot of the ladder is 6.5 metres from the house. The length of the ladder is
A kite is flying at a height of 50 m. If the length of the string is 100 m then the inclination of the string to the horizontal ground in degree measures is:
A. 90
B. 45
C. 60
D. 30
Two poles of the height 15 m and 20 m stand vertically upright on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.
A. 11 m
B. 12 m
C. 13 m
D. 14 m
