An aerialist on a high platform holds onto a trapeze attached to a support by an 7.9-m cord. (See the drawing.) Just before he jumps off the platform, the cord makes an angle of 37° with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is 0.57 m below its initial height. Calculate the angle θ that the trapeze cord makes with the vertical at this instant.

Question

An aerialist on a high platform holds onto a trapeze attached to a support by an 7.9-m cord. (See the drawing.) Just before he jumps off the platform, the cord makes an angle of 37° with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is 0.57 m below its initial height. Calculate the angle θ that the trapeze cord makes with the vertical at this instant.

Answer 1

An aerialist on a high platform holds onto a trapeze attached to a support by an 8.0-m cord. (See the drawing.) Just before he jumps off the platform, the cord makes an angle of 36° with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is 0.60 m below its initial height. Calculate the angle θ that the trapeze cord makes with the vertical at this instant.

Answer 2

To calculate the angle θ that the trapeze cord makes with the vertical at the instant it is 0.57 m below its initial height, we can use trigonometry.

Let's break down the problem step by step:

1. First, let's find the length of the cord when the aerialist jumps off the platform. We can use the given angle and the length of the cord to find the vertical component of the cord.

Using the given angle of 37° and the length of the cord (7.9 m), we can calculate the vertical component (h) of the cord using the equation:

h = 7.9 m * sin(37°)

2. Next, we need to find the height difference between the initial position and the release point. The aerialist swings down and then back up, releasing the trapeze when it is 0.57 m below its initial height. The initial height difference (Δh) can be calculated using the equation:

Δh = 0.57 m

3. Now, we can calculate the final height (H) of the trapeze at the release point. The final height can be found by subtracting the height difference from the initial height:

H = h - Δh

4. Finally, let's calculate the angle θ that the trapeze cord makes with the vertical at the release point. We can use the height difference (Δh) and the final height (H) to calculate the angle using the equation:

θ = arcsin(Δh / H)

Plug in the known values and calculate the angle θ.

Please note that if the final height (H) is zero or negative, it means that the trapeze would not reach that height, and there is no angle to calculate.

Using this method, you can find the angle θ that the trapeze cord makes with the vertical at the instant it is 0.57 m below its initial height.

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To calculate the angle θ that the trapeze cord makes with the vertical at the instant it is 0.57 m below its initial height, we can use trigonometry.