High School

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

Answer :

We are given the formula for the period of a pendulum:

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

with the value [tex]$T = 1.57$[/tex] seconds and [tex]$\pi = 3.14$[/tex].

Step 1. Isolate the square root term by dividing both sides by [tex]$2\pi$[/tex]:

[tex]$$
\sqrt{\frac{L}{32}} = \frac{T}{2\pi} = \frac{1.57}{2 \times 3.14}.
$$[/tex]

Step 2. Calculate the right-hand side:

[tex]$$
\frac{1.57}{6.28} = 0.25.
$$[/tex]

So we have:

[tex]$$
\sqrt{\frac{L}{32}} = 0.25.
$$[/tex]

Step 3. Square both sides of the equation to remove the square root:

[tex]$$
\frac{L}{32} = (0.25)^2 = 0.0625.
$$[/tex]

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

[tex]$$
L = 32 \times 0.0625 = 2.
$$[/tex]

Thus, the length of the pendulum is [tex]$2$[/tex] feet.

Final Answer: The length is [tex]$\boxed{2 \text{ feet}}$[/tex].