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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.
Solution :
Given:
- Height of the building ($BC$) = $20$ m.
- The transmission tower ($CD$) is fixed on top of the building.
- Angle of elevation from a point $A$ on the ground to the bottom of the tower (point $C$) = $45^\circ$.
- Angle of elevation from point $A$ on the ground to the top of the tower (point $D$) = $60^\circ$.
To find:
The height of the transmission tower ($h = CD$).
Step 1: Define variables and identify triangles.
Let $AB = x$ be the distance from the point on the ground to the base of the building. Let $CD = h$ be the height of the tower. The total height of the building and tower is $BD = BC + CD = 20 + h$.
Step 2: Analyze $\triangle ABC$ (Right-angled at $B$).
In $\triangle ABC$, the angle of elevation $\angle BAC = 45^\circ$.
Using the trigonometric ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
$\tan(45^\circ) = \frac{BC}{AB}$
Since $\tan(45^\circ) = 1$ [Trigonometric table value]:
$1 = \frac{20}{x}$
$x = 20$ m [Equation 1]
Step 3: Analyze $\triangle ABD$ (Right-angled at $B$).
In $\triangle ABD$, the angle of elevation $\angle BAD = 60^\circ$.
$\tan(60^\circ) = \frac{BD}{AB}$
Since $\tan(60^\circ) = \sqrt{3}$ [Trigonometric table value]:
$\sqrt{3} = \frac{20 + h}{x}$
Step 4: Solve for $h$.
Substitute $x = 20$ from Equation 1 into the equation from Step 3:
$\sqrt{3} = \frac{20 + h}{20}$
$20\sqrt{3} = 20 + h$ [Multiplying both sides by 20]
$h = 20\sqrt{3} - 20$
$h = 20(\sqrt{3} - 1)$
Using $\sqrt{3} \approx 1.732$:
$h = 20(1.732 - 1)$
$h = 20(0.732)$
$h = 14.64$ m
Final Answer: The height of the transmission tower is $20(\sqrt{3} - 1)$ m or approximately $14.64$ 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).
- 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.
- 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.
- 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.
- 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.
CBSE Solutions for Class 10 Mathematics Applications of Trigonometry
Chapters in CBSE - Class 10 Mathematics
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