A child is swinging at the end of a 3m rope. When the rope makes a 30 degree angle with the vertical, how high is she above her path's lowest point?

3m x (1 - cos 30)= 0.402 m

To solve this problem, we can break it down into components and use trigonometry. Here are the steps to calculate the height above the lowest point:

Step 1: Draw a diagram of the situation. Label the swing, the rope, and the angles involved.

Step 2: Identify the relevant angles and sides in the diagram. In this case, we have a right triangle formed by the rope, the vertical line, and the swing's path.

Step 3: Use trigonometry to find the height. Since we know the length of the rope (3m) and the angle it makes with the vertical (30 degrees), we can use the sine function to calculate the height.

sin(theta) = opposite/hypotenuse

In this case:

sin(30 degrees) = height/3m

Step 4: Rearrange the equation to solve for the height:

height = sin(30 degrees) * 3m

Step 5: Calculate the height using a calculator:

height = 0.5 * 3m = 1.5m

Therefore, when the rope makes a 30-degree angle with the vertical, the child is 1.5 meters above the lowest point on her path.

To find out how high the child is above her path's lowest point, we need to use basic trigonometry.

Let's visualize the situation. We have a child swinging on a rope, forming a right-angled triangle between the child, the lowest point of the swing's path, and the vertical line.

Here's how we can solve it step by step:

1. Draw a diagram: Draw a vertical line to represent the rope, and label the lowest point as "A." At the top of the swing, label the child as "B" and connect it to the lowest point with a straight line.

2. Identify the right-angled triangle: The right-angled triangle is formed between points A, B, and the vertical line.

3. Determine the angle: The problem states that the rope makes a 30-degree angle with the vertical. Label this angle as "θ."

4. Identify the sides of the triangle: The side opposite the angle θ is the height we want to find, so let's label it as "h." The side adjacent to θ is the horizontal distance from point B to the lowest point, which we can label as "x." The hypotenuse is the length of the rope, given as 3m.

5. Apply trigonometry: We will use the trigonometric function sine (sin) to relate the angle and the sides of the triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

In this case, sin(θ) = opposite/hypotenuse, so sin(30°) = h/3.

6. Solve for h: Rearrange the equation to solve for h. Multiply both sides by 3 to eliminate the fraction:

h = 3 * sin(30°).

7. Calculate the value: Evaluate sin(30°) using a calculator or table. The sine of 30 degrees is 0.5, so substituting the value gives:

h = 3 * 0.5 = 1.5m.

Therefore, when the rope makes a 30-degree angle with the vertical, the child is 1.5 meters above the lowest point of her swing's path.