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The question involves a pendulum's period formula: 

\[
P = 2\pi \sqrt{\frac{L}{32}}
\]

where:
- \( P \) is the period (time in seconds for the pendulum to swing back and forth),
- \( L \) is the pendulum's length,
- \( 32 \) represents the gravitational constant factor in this context.

The task is to explain how to solve for \( L \) given a value of \( P \), and identify the correct sequence of algebraic steps among the provided options.

To isolate \( L \), you would follow these steps:

1. **Square both sides** to eliminate the square root:
   \[
   P^2 = (2\pi)^2 \frac{L}{32}
   \]
2. **Divide both sides by \( (2\pi)^2 \)** to isolate \( \frac{L}{32} \):
   \[
   \frac{P^2}{(2\pi)^2} = \frac{L}{32}
   \]
3. **Multiply both sides by 32** to solve for \( L \):
   \[
   L = \frac{32 \cdot P^2}{(2\pi)^2}
   \]

Based on these steps, the correct answer should be one that follows this pattern: squaring both sides first, then dividing by \( (2\pi)^2 \), and finally multiplying by 32.

Would you like more details, or do you have any specific questions?

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### Related Questions
1. What is the formula for the period of a simple pendulum and what factors affect it?
2. Why is the gravitational constant represented as 32 in this equation?
3. How does changing the length \( L \) of a pendulum affect its period \( P \)?
4. What is the significance of squaring both sides in equations with square roots?
5. How can you rearrange other physics formulas to isolate specific variables?

### Tip
When isolating variables in square-root equations, squaring both sides is usually the first step to eliminate the root, making it easier to work with the remaining terms.

The time it takes a pendulum to swing back and forth is dependent on the length of the pendulum. The formula for this is P = 2π √(L/32), where P is the time in seconds it takes the pendulum to swing back and forth, L is the length in feet of the pendulum, and 32 represents the force due to gravity. Explain how the formula can be used to determine the length of the pendulum if the time of the swing is known.

Learn how to solve for the length of a pendulum given its period using the formula P = 2π √(L/32). This guide provides a step-by-step solution, covering algebraic manipulation and square root properties, suitable for high school physics or algebra students in Grades 9-11.

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