To find the length of the pendulum [tex]\( L \)[/tex] given the period [tex]\( T = 6.28 \)[/tex] seconds and [tex]\(\pi = 3.14\)[/tex], we need to use the given formula for the period of a pendulum [tex]\( T = 2\pi\sqrt{\frac{L}{22}} \)[/tex].
First, we start with the formula:
[tex]\[ T = 2\pi\sqrt{\frac{L}{22}} \][/tex]
Next, we substitute the known values [tex]\( T = 6.28 \)[/tex] and [tex]\(\pi = 3.14\)[/tex] into the formula:
[tex]\[ 6.28 = 2 \times 3.14 \times \sqrt{\frac{L}{22}} \][/tex]
Simplify the multiplication on the right-hand side:
[tex]\[ 6.28 = 6.28 \times \sqrt{\frac{L}{22}} \][/tex]
Now, isolate the square root term by dividing both sides of the equation by 6.28:
[tex]\[ 1 = \sqrt{\frac{L}{22}} \][/tex]
Next, square both sides of the equation to eliminate the square root:
[tex]\[ 1^2 = \left(\sqrt{\frac{L}{22}}\right)^2 \][/tex]
[tex]\[ 1 = \frac{L}{22} \][/tex]
Finally, solve for [tex]\( L \)[/tex] by multiplying both sides of the equation by 22:
[tex]\[ L = 22 \][/tex]
Therefore, the length [tex]\( L \)[/tex] of the pendulum is 22 feet. None of the provided options (32 feet, 3.2 feet, 6.4 feet, 64 feet) match the calculated pendulum length:
[tex]\[ \boxed{22 \text{ feet}} \][/tex]
