 An aerialist on a high platform holds on to 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 41° with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is 0.75 m below its initial height. Calculate the angle  0 that the trapeze cord makes with the vertical at this

instant 

To calculate the angle \(\theta\) that the trapeze cord makes with the vertical when the aerialist releases the trapeze, we can use some basic trigonometric relationships.

### Step 1: Determine the initial and final positions
- The initial length of the cord (\(L\)) is 8.0 meters.
- The trapeze initially makes an angle of 41° with the vertical.
- The final position of the aerialist is 0.75 meters below the initial height.

### Step 2: Calculate the vertical height of the aerialist's initial position
The vertical height of the aerialist above the lowest point (just before jumping) can be calculated using the cosine of the initial angle:

\[
h_{\text{initial}} = L \cdot \cos(41^\circ)
\]

### Step 3: Calculate the vertical height of the aerialist's final position
The final height is 0.75 meters lower than the initial height, so:

\[
h_{\text{final}} = h_{\text{initial}} - 0.75 \text{ meters}
\]

### Step 4: Use trigonometry to find the final angle \(\theta\)
Now, let's find the final angle \(\theta\) that the cord makes with the vertical. We use the fact that:

\[
\cos(\theta) = \frac{h_{\text{final}}}{L}
\]

### Step 5: Plug in the numbers and solve
1. Calculate \(h_{\text{initial}}\):

\[
h_{\text{initial}} = 8.0 \cdot \cos(41^\circ) 
\]
\[
h_{\text{initial}} \approx 8.0 \cdot 0.7547 \approx 6.0376 \text{ meters}
\]

2. Calculate \(h_{\text{final}}\):

\[
h_{\text{final}} = 6.0376 - 0.75 = 5.2876 \text{ meters}
\]

3. Find \(\cos(\theta)\):

\[
\cos(\theta) = \frac{5.2876}{8.0} \approx 0.66095
\]

4. Finally, find \(\theta\):

\[
\theta = \cos^{-1}(0.66095) \approx 48.58^\circ
\]

So, the angle \(\theta\) that the trapeze cord makes with the vertical when the aerialist releases it is approximately **48.58°**.

Would you like further clarification or have any questions? Here are some related questions:

1. How would the angle change if the aerialist dropped 1.0 meter instead of 0.75 meters?
2. Can you calculate the horizontal distance the aerialist traveled when releasing the trapeze?
3. What role does the length of the cord play in determining the angle?
4. How would the angle change if the initial angle was 30° instead of 41°?
5. What are the effects of gravity on the aerialist's motion during the swing?

**Tip:** Understanding how trigonometric functions relate to angles and sides of a triangle can help solve many physics problems involving motion and forces.

Learn how to calculate the angle that the trapeze cord makes with the vertical when an aerialist releases it. This problem involves trigonometry and kinematics, including the use of vertical motion equations. Suitable for high school students studying physics.

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