To solve for the length \( L \) of a pendulum that makes one full swing in 2.2 seconds using the given formula \( T = 2 \pi \sqrt{\frac{L}{32}} \), we need to follow these steps:
1. Substitute the given values into the formula \( T = 2 \pi \sqrt{\frac{L}{32}} \):
[tex]\[
2.2 = 2 \times 3.14 \times \sqrt{\frac{L}{32}}
\][/tex]
2. Isolate the square root term. First, divide both sides by \( 2 \times 3.14 \):
[tex]\[
\frac{2.2}{2 \times 3.14} = \sqrt{\frac{L}{32}}
\][/tex]
3. Calculate the value of the left-hand side:
[tex]\[
\frac{2.2}{6.28} \approx 0.35
\][/tex]
Thus,
[tex]\[
0.35 = \sqrt{\frac{L}{32}}
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
(0.35)^2 = \frac{L}{32}
\][/tex]
[tex]\[
0.1225 = \frac{L}{32}
\][/tex]
5. Isolate \( L \) by multiplying both sides by 32:
[tex]\[
L = 0.1225 \times 32 = 3.92
\][/tex]
The calculated value of \( L \) is approximately 3.92 feet.
Next, we compare this length to the given options (2 feet, 4 feet, 11 feet, 19 feet) to find the closest match:
- Subtract 3.92 from each option:
[tex]\[
|2 - 3.92| = 1.92
\][/tex]
[tex]\[
|4 - 3.92| = 0.08
\][/tex]
[tex]\[
|11 - 3.92| = 7.08
\][/tex]
[tex]\[
|19 - 3.92| = 15.08
\][/tex]
The option 4 feet has the smallest difference from 3.92 feet.
Therefore, the closest length of the pendulum that makes one full swing in 2.2 seconds is:
[tex]\[
\boxed{4 \text{ feet}}
\][/tex]
