To determine the length [tex]\( L \)[/tex] of the pendulum given the period [tex]\( T \)[/tex] and the value of [tex]\( \pi \)[/tex], follow these steps:
1. Given Equation:
[tex]\[
T = 2 \pi \sqrt{\left(\frac{L}{32}\right)}
\][/tex]
2. Substitute the Known Values:
[tex]\[
T = 9.42 \quad \text{and} \quad \pi = 3.14
\][/tex]
Thus, the equation becomes:
[tex]\[
9.42 = 2 \times 3.14 \times \sqrt{\left(\frac{L}{32}\right)}
\][/tex]
3. Isolate the Square Root Term:
Divide both sides by [tex]\( 2 \times 3.14 \)[/tex]:
[tex]\[
\frac{9.42}{2 \times 3.14} = \sqrt{\left(\frac{L}{32}\right)}
\][/tex]
4. Calculate the Value Inside the Square Root:
First, compute the denominator:
[tex]\[
2 \times 3.14 = 6.28
\][/tex]
Then,
[tex]\[
\frac{9.42}{6.28} = 1.5
\][/tex]
So,
[tex]\[
1.5 = \sqrt{\left(\frac{L}{32}\right)}
\][/tex]
5. Square Both Sides to Eliminate the Square Root:
[tex]\[
1.5^2 = \frac{L}{32}
\][/tex]
[tex]\[
2.25 = \frac{L}{32}
\][/tex]
6. Solve for [tex]\( L \)[/tex]:
Multiply both sides by 32:
[tex]\[
L = 32 \times 2.25
\][/tex]
[tex]\[
L = 72
\][/tex]
Therefore, the length of the pendulum is [tex]\( \boxed{72} \)[/tex] feet.
