To determine the length of each slide, we use trigonometry, specifically the cosine function, which relates the angle, the adjacent side (height of the slide), and the hypotenuse (length of the slide).
Slide for Younger Children: Height = 1.5 m, Angle = 30°. The cosine of 30° is √3/2. Using the formula cos(30°) = height/length, we get √3/2 = 1.5/length. Solving for the length, we find it to be 1.5 × 2/√3, approximately 1.73 meters. Slide for Older Children: Height = 3 m, Angle = 60°. The cosine of 60° is 1/2. Using cos(60°) = height/length, we get 1/2 = 3/length. Solving for the length, we find it to be 3 × 2, which is 6 meters.
Thus, the slide for younger children should be approximately 1.73 meters long, and for older children, it should be 6 meters long.

Let’s discuss in detail

## Trigonometric Applications in Playground Equipment

Trigonometry, a vital branch of mathematics, finds practical applications in designing playground equipment, such as slides. The challenge often lies in ensuring safety while providing an enjoyable experience for children of different age groups. A contractor tasked with installing two slides for different age groups in a park needs to consider the height and inclination of each slide. By applying trigonometric principles, specifically the cosine function, the contractor can determine the appropriate length for each slide, ensuring they are safe and suitable for the intended age groups.

### Design Considerations for Younger Children’s Slide

For children below the age of 5, safety is paramount. The contractor plans to install a slide with a top height of 1.5 meters, inclined at a 30° angle to the ground. This angle is gentle enough to ensure safety for younger children while still providing an enjoyable experience. The key is to calculate the correct length of the slide, which must be neither too steep nor too short, to maintain both the fun factor and safety.

#### Applying Trigonometry to the Younger Children’s Slide

To calculate the length of the slide for younger children, we use the cosine function. The cosine of an angle in a right-angled triangle is the ratio of the adjacent side (height of the slide) to the hypotenuse (length of the slide). For a slide height of 1.5 meters and an angle of 30°, the formula becomes cos(30°) = 1.5/(length of the slide). The cosine of 30° is known to be √3/2.

Calculating the Length of the Younger Children’s Slide
Solving the equation √3/2 = 1.5/length for the length of the slide, we find that the length is 1.5 × 2/√3, which simplifies to approximately 1.73 meters. This length ensures that the slide is long enough for a gentle descent, suitable for younger children, balancing enjoyment with safety.

##### Design and Safety for the Older Children’s Slide

For older children, the contractor plans a steeper slide, with a top height of 3 meters and an inclination of 60°. This steeper angle provides more excitement suitable for older children, who can safely handle a more thrilling descent. However, it’s crucial to calculate the correct length of the slide to ensure it is not excessively steep, which could compromise safety.

Calculating the Length of the Older Children’s Slide
Applying the cosine function again, we use the formula cos(60°) = 3/(length of the slide). The cosine of 60° is 1/2. Solving the equation 1/2 = 3/length for the length, we find that the length is 3×2, which equals 6 meters. This length provides a thrilling yet safe experience for older children, offering a steeper slide that is appropriately proportioned for their age and ability.

###### Trigonometry’s Role in Playground Safety and Design

These calculations illustrate the practical application of trigonometry in designing playground equipment. By using trigonometric functions, the contractor can determine the optimal lengths for slides, ensuring they are both fun and safe for children of different ages. This approach highlights how mathematical principles are crucial in real-world applications, particularly in areas where safety and enjoyment are key considerations.

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.