To find out how long the swing is using the pendulum formula, we start with the given formula:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
Given:
- [tex]\( T \)[/tex] (the period) is 3.1 seconds.
- The acceleration due to gravity [tex]\( g \)[/tex] is 32 ft/s[tex]\(^2\)[/tex].
The formula needs to be rearranged to solve for [tex]\( L \)[/tex]:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
First, isolate [tex]\( \sqrt{\frac{L}{32}} \)[/tex] on one side by dividing both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[
\frac{T}{2 \pi} = \sqrt{\frac{L}{32}}
\][/tex]
Next, square both sides to eliminate the square root:
[tex]\[
\left(\frac{T}{2 \pi}\right)^2 = \frac{L}{32}
\][/tex]
Then, multiply both sides by 32 to solve for [tex]\( L \)[/tex]:
[tex]\[
L = 32 \left(\frac{T}{2 \pi}\right)^2
\][/tex]
Substitute [tex]\( T = 3.1 \)[/tex]:
[tex]\[
L = 32 \left(\frac{3.1}{2 \pi}\right)^2
\][/tex]
Calculate the value inside the parentheses first:
[tex]\[
\frac{3.1}{2 \pi} \approx 0.493
\][/tex]
Then, square this value:
[tex]\[
(0.493)^2 \approx 0.243
\][/tex]
Finally, multiply by 32:
[tex]\[
L = 32 \times 0.243 \approx 7.78957259842293
\][/tex]
Rounding 7.78957259842293 to the tenths place:
[tex]\[
L \approx 7.8
\][/tex]
Therefore, the length of the swing is approximately:
[tex]\[
\boxed{7.8 \text{ feet}}
\][/tex]
So, the correct choice is:
B. 7.8 feet
