Certainly! Let's break this down step by step to compute the value of [tex]\( 2 \pi \sqrt{\frac{L}{g}} \)[/tex] when [tex]\(\pi = 3 \frac{1}{7}\)[/tex], [tex]\(L = 98\)[/tex], and [tex]\(g = 32\)[/tex].
1. Express [tex]\(\pi\)[/tex] as an improper fraction:
Given:
[tex]\[
\pi = 3 \frac{1}{7}
\][/tex]
To convert this mixed number into an improper fraction:
[tex]\[
\pi = 3 + \frac{1}{7} = \frac{3 \times 7 + 1}{7} = \frac{21 + 1}{7} = \frac{22}{7}
\][/tex]
2. Substitute the values:
We need to find the value of [tex]\( 2 \pi \sqrt{\frac{L}{g}} \)[/tex] using the given values:
[tex]\[
\pi = \frac{22}{7}, \quad L = 98, \quad g = 32
\][/tex]
3. Calculate [tex]\(\sqrt{\frac{L}{g}}\)[/tex]:
First, compute the ratio [tex]\(\frac{L}{g}\)[/tex]:
[tex]\[
\frac{L}{g} = \frac{98}{32}
\][/tex]
Simplify this fraction:
[tex]\[
\frac{98}{32} = \frac{49}{16}
\][/tex]
Now, take the square root of [tex]\(\frac{49}{16}\)[/tex]:
[tex]\[
\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4}
\][/tex]
4. Substitute and compute [tex]\( 2 \pi \sqrt{\frac{L}{g}} \)[/tex]:
Now, replace [tex]\(\pi\)[/tex] and [tex]\(\sqrt{\frac{L}{g}}\)[/tex] in the expression:
[tex]\[
2 \pi \sqrt{\frac{L}{g}} = 2 \times \frac{22}{7} \times \frac{7}{4}
\][/tex]
Simplify the expression:
[tex]\[
2 \times \frac{22}{7} \times \frac{7}{4} = 2 \times \frac{22 \times 7}{7 \times 4} = 2 \times \frac{22}{4} = 2 \times 5.5 = 11
\][/tex]
Thus, the value of [tex]\( 2 \pi \sqrt{\frac{L}{g}} \)[/tex] is [tex]\( \boxed{11} \)[/tex].
