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This problem involves a wrecking ball suspended by two strings. We are asked to determine the tension in string \( B \) before it is cut, with the given angles and mass.

### Given:
- Mass of wrecking ball, \( M = 80.0 \, \text{kg} \)
- Angles: 
  - \( \theta_A = 30^\circ \) (Angle between string \( A \) and the ceiling)
  - \( \theta_B = 45^\circ \) (Angle between string \( B \) and the ceiling)
  
The wrecking ball is in equilibrium, so the forces in both the horizontal and vertical directions must balance.

### Step-by-Step Approach:
1. **Force Equilibrium in the Vertical Direction:**
   The vertical components of the tensions in strings \( A \) and \( B \) must balance the weight of the wrecking ball:
   \[
   T_A \sin(30^\circ) + T_B \sin(45^\circ) = M g
   \]
   where \( g \approx 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity.

2. **Force Equilibrium in the Horizontal Direction:**
   The horizontal components of the tensions in strings \( A \) and \( B \) must balance each other:
   \[
   T_A \cos(30^\circ) = T_B \cos(45^\circ)
   \]

### Solving the System of Equations:
First, we solve the horizontal equation for \( T_A \) in terms of \( T_B \):
\[
T_A = T_B \frac{\cos(45^\circ)}{\cos(30^\circ)}
\]

Now, substitute this expression for \( T_A \) into the vertical equilibrium equation:
\[
T_B \frac{\cos(45^\circ)}{\cos(30^\circ)} \sin(30^\circ) + T_B \sin(45^\circ) = M g
\]

Now, I'll solve this equation to find \( T_B \), the tension in the string \( B \).

Let me compute that.The tension in string \( B \) (the one Miley is about to cut) is approximately **702.92 N**.

The tension in string \( A \) is approximately **573.93 N**.

Would you like further details or clarifications?

### Related questions:
1. How would the tensions change if the mass of the wrecking ball increased?
2. What would happen if both angles were equal?
3. How does the angle between the strings affect the tension?
4. How could this system be solved graphically?
5. What forces are acting on the wrecking ball in equilibrium?

### Tip:
Always break forces into components (horizontal and vertical) when dealing with systems in equilibrium, as this helps simplify the problem into solvable equations.

Miley Cyrus spots a wrecking ball of mass M = 80.0 [kg] that is suspended by two strings as shown. She wants to cut string B so that she can swing on it. What is the tension on the string that she's about to cut?

Determine the tension in the string supporting a wrecking ball of mass 80 kg suspended by two strings at angles of 30° and 45°. This problem involves trigonometric concepts and the equilibrium of forces, suitable for students in Grades 11-12.

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