To find the height of the poles and the distances from the point to each pole, we use trigonometry. Let’s denote the height of the poles as h and the distances from the point to the poles as d1 and d2.
For the Pole with a 60° Angle of Elevation: Using tan(60°) = √3, the equation is √3 = h/d1.
For the Pole with a 30° Angle of Elevation: Using tan(30°) = 1/√3, the equation is 1/√3 = h/d2.
Since the total width of the road is 80 m, d1 + d2 = 80 m. Solving these equations simultaneously, we find h = 40√3 m approximately 69.28 m), d1 = 40, and d2 = 40 m. Therefore, the height of the poles is about 69.28 meters, and the distances from the point to each pole are both 40 meters.

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## Trigonometric Applications in Urban Planning

Trigonometry, an essential branch of mathematics, is often applied in urban planning and design, particularly in situations involving measurements and spatial relationships. A common application is determining the heights of structures and distances between them when direct measurement is not feasible. The problem at hand involves two poles of equal height on opposite sides of an 80-meter-wide road. From a point on the road, the angles of elevation to the tops of these poles are given. This scenario is a practical demonstration of how trigonometry can be used to solve real-world problems in urban and civil engineering.

### Understanding the Problem: Two Poles and a Road

The problem presents two poles of equal height standing on either side of a road that is 80 meters wide. From a point on the road, the angles of elevation to the top of the poles are 60° and 30°, respectively. The objective is to determine the height of the poles and the distances from the point to each pole. This setup forms two right-angled triangles, one with each pole, sharing a common horizontal distance (the width of the road).

#### 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 pole) to the length of the adjacent side (distance from the point to the pole). By applying the tangent function to the given angles of elevation, we can calculate the heights of the poles and the distances from the point to each pole.

Calculating the Distances to the Poles
First, we use the tangent function for the 60° angle of elevation to find the distance to one pole. The equation is tan(60°) = √3 = h/d1, where d1 is the distance to the first pole. Similarly, for the 30° angle of elevation to the other pole, the equation is tan(30°) = 1/√3 = h/d2, where d2 is the distance to the second pole.

##### Solving for the Height and Distances

Since the total width of the road is 80 meters, we have d1 + d2 = 80 meters. Solving the equations √3 = h/d1 and 1/√3 = h/d2 simultaneously with d1 + d2 = 80 meters, we find the height of the poles and the distances from the point to each pole.

###### Practical Implications of Trigonometry in Measurement

The solution reveals that the height of the poles is approximately 69.28 meters, and the distances from the point to each pole are both 40 meters. This example illustrates the practical application of trigonometry in urban and civil engineering, demonstrating how it can provide accurate measurements in complex scenarios. Trigonometry proves to be an invaluable tool in urban planning, offering a mathematical approach to spatial problem-solving where direct measurement is not possible.

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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.