To find the height of the TV tower and the width of the canal, we use trigonometry. Let’s denote the height of the tower as h and the width of the canal as d.
From the Point Directly Opposite (60° Angle of Elevation): Using tan(60°) = √3, the equation is √3 = h/d.
From the Point 20 m Away (30° Angle of Elevation): The distance from this point to the tower’s foot is d + 20 m. Using tan(30°) = 1/√3, the equation is 1/√3 = h/(d+20).
Solving these equations simultaneously, we find h = 20√3 m (approximately 34.64 m) and d = 20 m. Therefore, the height of the tower is about 34.64 meters, and the width of the canal is 20 meters.

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## Trigonometric Applications in Geometric Measurement

Trigonometry, a vital branch of mathematics, is extensively used in geometric measurement, particularly in determining the dimensions of structures and distances in scenarios where direct measurement is not feasible. The problem at hand involves a TV tower standing on the bank of a canal and requires determining the tower’s height and the canal’s width. This is achieved by analyzing the angles of elevation from two different points on the opposite bank. This scenario exemplifies the practical application of trigonometry in real-world situations, demonstrating its importance in fields like engineering, surveying, and architecture.

### Understanding the Problem: TV Tower and Canal

The problem presents a TV tower standing vertically on one bank of a canal. From a point directly opposite the tower on the other bank, the angle of elevation to the top of the tower is 60°. From another point 20 meters away from the first, along the line joining this point to the foot of the tower, the angle of elevation is 30°. The objective is to determine the height of the tower and the width of the canal, forming two distinct right-angled triangles with the tower.

#### The Role of Tangent in Angle of Elevation

In trigonometry, the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side (height of the tower) to the length of the adjacent side (horizontal distance from the point to the tower). By applying the tangent function to the given angles of elevation, we can calculate the height of the tower and the width of the canal.

Calculating the Width of the Canal
First, we use the tangent function for the 60° angle of elevation from the point directly opposite the tower. The equation is tan(60°) = √3 = h/d, where d is the width of the canal. This equation will be used in conjunction with the second angle of elevation to solve for h and d.

##### Determining the Height of the Tower

Next, we apply the tangent function to the 30° angle of elevation from the point 20 meters away. The equation is tan(30°) = 1/√ = h/(d+20). This equation, combined with the first, allows us to solve for the height of the tower and the width of the canal simultaneously.

###### Trigonometry in Practical Measurement

Solving the equations √3 = h/d and 1/√3 = h/(d+20)simultaneously, we find that the height of the tower is approximately 34.64 meters, and the width of the canal is 20 meters. This example illustrates the practical application of trigonometry in determining dimensions and distances in real-life scenarios. Trigonometry proves to be an invaluable tool in various fields, offering a mathematical approach to solving problems where direct measurement is challenging or impossible.

Discuss this question in detail or visit to Class 10 Maths Chapter 9 for all questions.
Questions of 10th Maths Exercise 9.1 in Detail

 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°. 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. 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 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case? 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. 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. 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. 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. 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. 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. 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. 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 joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal. 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. 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. 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°. Find the distance travelled by the balloon during the interval. 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.