High School

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Hamlet wants to try out acrobatics. He stands on a high platform and holds on to a trapeze attached to a support cord of length 9.1 m. Just before he jumps off the platform, the cord makes an angle of 39.5 degrees with the vertical. He jumps, swings down, then up, releasing the trapeze at the instant it is 0.90 m below its initial height.

Calculate the angle θ that the trapeze cord makes with the vertical at this instant. Enter your answer in degrees.

Answer :

To solve this problem, we need to determine the angle [tex]\theta[/tex] that the trapeze cord makes with the vertical after Hamlet releases it when it is 0.90 meters below its initial height.


  1. Initial Setup and Triangle Consideration:


    • Initially, the trapeze cord makes an angle of [tex]39.5^\circ[/tex] with the vertical.

    • The length of the trapeze cord is given as 9.1 meters.

    • After Hamlet swings and releases, the trapeze is 0.90 meters lower than its original height.



  2. Calculate the Change in Vertical Position:


    • When the trapeze is released 0.90 meters lower, the vertical change from the initial position is given as:
      [tex]h = 0.90 \text{ meters}[/tex]

    • This means the new vertical position of the trapeze above the horizontal line from the platform is:
      [tex]y = (9.1 - 0.90) = 8.2 \text{ meters}[/tex]



  3. Determine the Angle [tex]\theta[/tex]:


    • Using the geometry of the situation, the cosine rule in a right triangle formed by the cord, and the vertical drop:

    • [tex]\cos(\theta) = \frac{8.2}{9.1}[/tex]

    • So we calculate:
      [tex]\cos(\theta) = \frac{8.2}{9.1} = 0.9011[/tex]

    • Now, to find the angle [tex]\theta[/tex]:
      [tex]\theta = \cos^{-1}(0.9011) \approx 25.7^\circ[/tex]




Conclusion: The angle [tex]\theta[/tex] that the trapeze cord makes with the vertical when Hamlet releases it is approximately [tex]25.7^\circ[/tex]. This detailed solution shows us how the physical condition changes the angle as the trapeze swings down and up before being released.

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Rewritten by : Barada