The period [tex]$ T $[/tex] (in seconds) of a pendulum is given by [tex]$ T = 2 \pi \sqrt{\frac{L}{32}} $[/tex], where [tex]$ L $[/tex] stands for the length (in feet) of the pendulum. If [tex]$\pi = 3.14$[/tex], and the period is 1.57 seconds, what is the length?

A. 16 feet
B. 8 feet
C. 20 feet
D. 2 feet

To find the length [tex]$ L $[/tex] of the pendulum when the period [tex]$ T $[/tex] is 1.57 seconds, we need to use the formula for the period of a pendulum:

[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]

We are given:
- [tex]$ T = 1.57 $[/tex] seconds
- [tex]$\pi = 3.14$[/tex]

Let's rearrange the formula to solve for [tex]$ L $[/tex].

1. Start with the formula:

[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]

2. To isolate the square root, divide both sides by [tex]$ 2 \pi $[/tex]:

[tex]\[ \frac{T}{2 \pi} = \sqrt{\frac{L}{32}} \][/tex]

3. Square both sides to remove the square root:

[tex]\[ \left(\frac{T}{2 \pi}\right)^2 = \frac{L}{32} \][/tex]

4. Multiply both sides by 32 to solve for [tex]$ L $[/tex]:

[tex]\[ L = 32 \left(\frac{T}{2 \pi}\right)^2 \][/tex]

Now, plug in the values:

- [tex]$ T = 1.57 $[/tex]
- [tex]$\pi = 3.14$[/tex]

[tex]\[ L = 32 \left(\frac{1.57}{2 \times 3.14}\right)^2 \][/tex]

Calculating the above expression results in:

[tex]\[ L \approx 2.0 \text{ feet} \][/tex]

Therefore, the length [tex]$ L $[/tex] of the pendulum that gives a period of 1.57 seconds is approximately 2 feet.