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Find the value of [tex][tex]$2 \pi \sqrt{\frac{L}{g}}$[/tex][/tex] when [tex]\pi = 3 \frac{1}{7}[/tex], [tex]L = 98[/tex], and [tex]g = 32[/tex].

Sagot :

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].