An aerialist on a high platform holds onto a trapeze attached to a support by an 9.8-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 0.77 m below its initial height. Calculate the angle θ that the trapeze cord makes with the vertical at this instant.
36.9
To find the angle θ that the trapeze cord makes with the vertical at the instant the aerialist releases it, we need to use trigonometric functions and the given information.
Let's break down the problem and find the solution step by step:
Step 1: Identify the relevant information.
- The trapeze cord has a length of 9.8 m.
- Just before jumping, the cord makes an angle of 37° with the vertical.
- The trapeze swings down and up, reaching a point 0.77 m below its initial height.
Step 2: Draw a diagram.
Sketch a diagram of the situation, labeling the relevant quantities. This will help us visualize the problem and better understand the given information.
```
|
|\
D | \
|_\
| \
| \
|Θ \
| \
|______\
| \
A | \ C
| \
```
In the above diagram:
- Point A represents the aerialist's initial position.
- Point B represents the lowest point of the swing.
- Point C represents the position where the trapeze is released.
- Point D represents the initial vertical position.
Step 3: Break down the problem into components.
Divide the problem into horizontal and vertical components, as this will help us apply trigonometric functions.
Let's consider the displacement of the trapeze from point A to point C:
- The vertical component of the displacement is 0.77 m.
- The horizontal component of the displacement is the difference between the initial vertical position and point C.
Step 4: Find the horizontal component of the displacement.
Since we know the length of the trapeze cord (9.8 m) and the angle it makes with the vertical (37°), we can use trigonometric functions to find the horizontal component of the displacement.
The horizontal component can be calculated using the formula:
Horizontal component = Length of cord * cos(angle)
Plugging in the values:
Horizontal component = 9.8 m * cos(37°)
Step 5: Calculate the horizontal displacement from D to C.
The horizontal displacement from point D to point C is the difference between the initial position (point D) and the horizontal component.
Horizontal displacement = 0 - Horizontal component
Step 6: Find the angle θ.
The angle θ can be obtained by finding the inverse tangent of the vertical displacement divided by horizontal displacement.
θ = arctan(Vertical displacement / Horizontal displacement)
Plugging in the values we calculated:
θ = arctan(0.77 m / (0 - Horizontal component))
Step 7: Calculate the final answer.
Using a scientific calculator, compute the arctan of the quantity calculated in step 6.
The resulting angle θ represents the angle that the trapeze cord makes with the vertical at the instant the aerialist releases it.
Note: Make sure to check the mode of your calculator (degrees or radians) and use the appropriate mode for your calculation.
By following these steps, you should be able to calculate the angle θ.