Tree Height Calculation Using Pythagorean Theorem
The problem describes a scenario that can be modeled using a right-angled triangle.
- The height at which the tree broke forms one leg of the triangle (let's call it a).
- The distance from the foot of the tree to where the broken part touches the ground forms the other leg (let's call it b).
- The broken upper part of the tree forms the hypotenuse (let's call it c).
Given values:
- Standing part of the tree, a = 8 m
- Distance on the ground, b = 6 m
Finding the Broken Part Length
We use the Pythagorean theorem to find the length of the broken part (hypotenuse, c):
$ a^2 + b^2 = c^2 $
Substitute the given values:
$ 8^2 + 6^2 = c^2 $
$ 64 + 36 = c^2 $
$ 100 = c^2 $
Solve for c:
$ c = \sqrt{100} $
$ c = 10 \text{ m} $
So, the length of the broken upper part is 10 m.
Calculating Total Tree Height
The total height of the tree is the sum of the standing part and the broken part.
Total Height = Standing Part (a) + Broken Part (c)
Total Height = 8 m + 10 m
Total Height = 18 m
The height of the tree is 18 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:
- 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.
- 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
