To solve this problem, we'll approach it by considering energy conservation and the geometry of the trapeze cord.

Step 1: Use Energy Conservation

We know that the aerialist swings down from a certain height and reaches a point where he is 0.71 meters below his initial height. We can use energy conservation to calculate the speed at this point, but we don't need the speed to determine the angle.

The critical point is that the change in height corresponds to a change in potential energy, and that change in height will also determine the angle that the trapeze cord makes with the vertical.

Step 2: Find the Change in Height

Let the initial height be denoted as $h_0$, and the height at the point where the aerialist releases the trapeze be $h_1$. We are given that the difference in height is:

$h_0 - h_1 = 0.71 \text{ meters}.$

Step 3: Determine the Geometry of the Problem

Initially, the trapeze cord makes a 42-degree angle with the vertical. The length of the cord is 6.7 meters. When the aerialist is at the lowest point in his swing, the cord is vertical, and he is at his maximum kinetic energy.

Now, to find the angle at the point where the aerialist is 0.71 meters below the initial height, we use the geometry of the situation.

• The length of the trapeze cord $L$ is 6.7 meters.
• The height difference is $\Delta h = 0.71 \text{ meters}$.

At this new position, the height of the aerialist relative to the original position is:

$h = L - \Delta h = 6.7 - 0.71 = 5.99 \text{ meters}.$

This is the new vertical distance from the point where the cord is attached to the aerialist.

Step 4: Use Trigonometry to Find the Angle

Now, to find the angle that the trapeze cord makes with the vertical, we can use the cosine of the angle $\theta$:

$\cos(\theta) = \frac{\text{vertical height (5.99 meters)}}{\text{length of the cord (6.7 meters)}}.$

$\cos(\theta) = \frac{5.99}{6.7}.$

Now, let's calculate $\theta$:

$\cos(\theta) = \frac{5.99}{6.7} \approx 0.8933.$

Taking the arccosine of this value:

$\theta = \cos^{-1}(0.8933).$

Step 5: Calculate the Angle

Let me calculate the angle for you.The angle that the trapeze cord makes with the vertical at the instant when the aerialist is 0.71 meters below the initial height is approximately 26.6 degrees.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How would the angle change if the cord was longer?
2. What if the height difference was larger—how would that affect the angle?
3. How does air resistance affect the aerialist's swing?
4. Could you calculate the aerialist's speed at the lowest point of the swing?
5. How would the calculation change if the aerialist swings at a different angle initially?

Tip: When analyzing physical problems involving swings, energy conservation often provides a powerful way to connect height differences with speed and angles.