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Q2:
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30°$ with it. The distance between the foot of the tree to the point where the top touches the ground is $8$ m. Find the height of the tree.
Solution :
Given:
- A tree breaks at a certain point, and the top touches the ground.
- The angle of elevation of the top of the tree from the point where it touches the ground is $\theta = 30^\circ$.
- The distance between the foot of the tree and the point where the top touches the ground is $BC = 8$ m.
To Find:
The total height of the tree ($H = AB + AC$, where $A$ is the top, $B$ is the foot, and $C$ is the point where the top touches the ground, with the break occurring at point $D$).
Step 1: Define the variables and model the triangle.
Let the original tree be $AD$. Let it break at point $D$. The part $AD$ bends such that the top $A$ touches the ground at point $C$. Let $B$ be the foot of the tree. Thus, $BD$ is the vertical part remaining standing, and $DC$ is the broken part that touches the ground. The total height of the tree is $H = BD + DC$.
In the right-angled triangle $\triangle DBC$ (right-angled at $B$):
- Base $BC = 8$ m
- Angle $\angle DCB = 30^\circ$
Step 2: Calculate the height of the standing part ($BD$).
Using the trigonometric ratio for tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
$\tan(30^\circ) = \frac{BD}{BC}$
Since $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ [Standard trigonometric value]:
$\frac{1}{\sqrt{3}} = \frac{BD}{8}$
$BD = \frac{8}{\sqrt{3}}$ m
Step 3: Calculate the length of the broken part ($DC$).
Using the trigonometric ratio for cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\cos(30^\circ) = \frac{BC}{DC}$
Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ [Standard trigonometric value]:
$\frac{\sqrt{3}}{2} = \frac{8}{DC}$
$DC = \frac{16}{\sqrt{3}}$ m
Step 4: Calculate the total height of the tree ($H$).
$H = BD + DC$
$H = \frac{8}{\sqrt{3}} + \frac{16}{\sqrt{3}}$
$H = \frac{24}{\sqrt{3}}$
Rationalizing the denominator by multiplying the numerator and denominator by $\sqrt{3}$:
$H = \frac{24 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{24\sqrt{3}}{3} = 8\sqrt{3}$ m
Using $\sqrt{3} \approx 1.732$:
$H = 8 \times 1.732 = 13.856$ m
Final Answer: The height of the tree is $8\sqrt{3}$ m (or approximately $13.86$ m).
More Questions from Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1
- Q1: A circus artist is climbing a $20$ m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is $30°$ (see Fig. 9.11).
- Q10: Two poles of equal heights are standing opposite 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 the poles are $60°$ and $30°$, respectively. Find the height of the poles and the distances of the point from the poles.
- Q11: A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60°$. From another point $20$ m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is $30°$ (see Fig. 9.12). Find the height of the tower and the width of the canal.
- Q12: From the top of a $7$ m high building, the angle of elevation of the top of a cable tower is $60°$ and the angle of depression of its foot is $45°$. Determine the height of the tower.
- Q13: As observed from the top of a $75$ m high lighthouse from the sea-level, the angles of depression of two ships are $30°$ and $45°$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
- Q14: A $1.2$ m tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2$ m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60°$. After some time, the angle of elevation reduces to $30°$ (see Fig. 9.13). Find the distance travelled by the balloon during the interval.
- Q15: A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30°$, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60°$. Find the time taken by the car to reach the foot of the tower from this point.
- Q3: A contractor plans to install two slides for the children to play in a park. For the children below the age of $5$ years, she prefers to have a slide whose top is at a height of $1.5$ m, and is inclined at an angle of $30°$ to the ground, whereas for elder children, she wants to have a steep slide at a height of $3$m, and inclined at an angle of $60°$ to the ground. What should be the length of the slide in each case?
- Q4: The angle of elevation of the top of a tower from a point on the ground, which is $30$ m away from the foot of the tower, is $30°$. Find the height of the tower.
- Q5: A kite is flying at a height of $60$ m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60°$. Find the length of the string, assuming that there is no slack in the string.
- Q6: A $1.5$ m tall boy is standing at some distance from a $30$ m tall building. The angle of elevation from his eyes to the top of the building increases from $30°$ to $60°$ as he walks towards the building. Find the distance he walked towards the building.
- Q7: From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a $20$ m high building are $45°$ and $60°$ respectively. Find the height of the tower.
- Q8: A statue, $1.6$ m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.
- Q9: The angle of elevation of the top of a building from the foot of the tower is $30°$ and the angle of elevation of the top of the tower from the foot of the building is $60°$. If the tower is $50$ m high, find the height of the building.
CBSE Solutions for Class 10 Mathematics Applications of Trigonometry
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