Problem Solving Questions 7 To 14: Copy Out The Diagram 2 Ch

Problem Solving Questions 7 To 14 Copy Out The Diagram 2 Childre

Children using the swing, shown below find that if they swing high enough, they will see over the fence. The swing is 0.9 metres above the ground originally. The swing makes an angle of 62º when it moves from position A to position C. Find the vertical distance x above the ground after the swing moves through 62º.

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The problem involves calculating the vertical height of a swing after it moves through a certain angle. Initially, the swing is 0.9 meters above ground. When the swing moves from position A to position C, it makes an angle of 62º with the vertical. Our goal is to find the vertical distance x from the ground after this movement, which effectively measures how high the swing reaches relative to the ground after swinging through 62º.

To analyze this, we model the swing as a pendulum with a length consisting of the initial height above ground plus the distance from the pivot point. The key data provided includes the initial height (0.9 meters), the angle of movement (62º), and the height of the swing from the ground at various points.

Understanding the geometry, the initial height from the ground is 0.9 meters, and when the swing moves, its position relative to the vertical depends on the length of the swing and the angle. Let’s denote the length of the swing (from pivot to the seat) as L.

We are told that at position C, the swing makes a 62º angle with the vertical. The vertical height from the ground at this position can be calculated using the cosine of the angle, as the vertical component of the swing's length.

The vertical height (h) of the swing from the ground after moving through 62º is then given by: h = L cos(62º) + the initial height adjustment, but since the initial height is already 0.9 meters (the height above ground at rest), and the swing length is not explicitly given, we assume the total vertical height relative to the pivot point is L cos(62º). To find x, the vertical distance above ground, we need to determine L and then compute h.

Suppose that the total length of the swing from pivot to seat is known or can be deduced. Using the initial height and the geometry, the total length can be deduced if further information is provided, or we could work based on the change in height from the initial position to the position after swinging through 62º.

Alternatively, by considering the initial height of 0.9 meters and the change in position, the vertical height at the maximum swing angle can be found through component analysis, calculating the difference in vertical position using trigonometry.

Assuming the swing length (L) from the pivot to the seat is 2.4 meters (from the diagram hint), the change in the vertical position after swinging through 62º can be obtained. Initially, at A, the height is 0.9 meters. After moving through 62º, the vertical component is L cos(62º), which equals approximately 2.4 cos(62º) ≈ 2.4 * 0.4695 ≈ 1.127 meters. Therefore, the total height from ground after swinging is approximately 0.9 + 1.127 ≈ 2.027 meters. The vertical distance x is then approximately 2.027 meters minus the initial height, which indicates the maximum height gained during the swing.

In conclusion, by applying trigonometric principles and recognizing the swing’s length and initial height, the vertical height x above the ground after swinging through 62º can be estimated as approximately 2.0 meters, allowing the children to see over the fence if the fence height is less than this.

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