To solve this problem, we use trigonometry, specifically the tangent function. Initially, the angle of elevation is 30°, and finally, it becomes 60°. Let’s denote the initial and final distances from the boy to the building as d1 and d2 respectively.
Initial Position (30° angle): Using tan(30°) = 1/√3, the height of the building minus the height of the boy is 30 m − 1.5 m = 28.5 m. So, 1/√3 = 28.5/d1. Solving for d1, we get d1 = 28.5√3.
Final Position (60° angle): Using tan(60°) = √3, we have √3 = 28.5/d2. Solving for d2, we get d2 = 28.5/√3.
​The distance walked is d1 − d2 = 28.5√3 − 28.5/√3, which simplifies to approximately 24.75 meters. Therefore, the boy walked about 24.75 meters towards the building.

Let’s discuss in detail

## Trigonometric Motion Analysis

Trigonometry, a branch of mathematics, is not only about solving theoretical problems but also about understanding real-world situations, such as the movement of a person in relation to a structure. The problem at hand involves a boy walking towards a building and the consequent change in the angle of elevation from his eyes to the top of the building. This scenario is a classic example of how trigonometry can be applied to analyze motion and distance in a physical context, demonstrating the practical utility of mathematical concepts in everyday life.

### Motion Towards a Building

The problem presents a 1.5-meter-tall boy standing at a certain distance from a 30-meter-tall building. As the boy walks towards the building, the angle of elevation from his eye level to the top of the building increases from 30° to 60°. The objective is to calculate the distance the boy has walked towards the building. This scenario forms two different right-angled triangles at different instances – one at the initial position (30° angle) and the other at the final position (60° angle).

#### 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 opposite side (in this case, the difference in height between the boy’s eyes and the top of the building) to the adjacent side (the distance from the boy to the building). By applying the tangent function to the angles of elevation, we can determine the boy’s distances from the building at both the initial and final positions.

Calculating the Initial Distance
Initially, when the angle of elevation is 30°, we use the formula tan(30°) = opposite/adjacent. The opposite side is the height difference between the building and the boy’s eyes, which is 30 m − 1.5 m = 28.5 m. The tangent of 30° is 1/√3. Therefore, 1/√3 = 28.5/d1, where d1 is the initial distance. Solving for d1, we find d1 = 28.5√3 meters.

##### Calculating the Final Distance

At the final position, where the angle of elevation is 60°, we apply the formula tan(60°) = opposite/adjacent. The tangent of 60° is √3. Therefore, √3 = 28.5/d2, where d2 is the final distance. Solving for d2, we find d2 = 28.5/√3 meters.

###### Determining the Distance Walked

The distance the boy walked towards the building is the difference between the initial and final distances, d1 − d2. Substituting the values, we get 28.5√ − 28.5/√3, which simplifies to approximately 24.75 meters. This calculation not only demonstrates the application of trigonometry in analyzing real-world motion but also highlights the importance of understanding mathematical principles to interpret and solve practical problems. The scenario underscores how trigonometry can provide insights into everyday situations, bridging the gap between abstract mathematics and tangible experiences.

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.