Vaughn Frazier
08/02/2023 · Primary School
\( \square < \sqrt { 112 } < \square \)
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Tutor-Verified Answer
Step-by-step Solution
To find the values that fit the inequality \( \square < \sqrt{112} < \square \), we first need to calculate \( \sqrt{112} \).
We can simplify \( \sqrt{112} \):
\[
\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16} \times \sqrt{7} = 4\sqrt{7}
\]
Next, we need to approximate \( \sqrt{7} \). We know that:
\[
\sqrt{4} = 2 \quad \text{and} \quad \sqrt{9} = 3
\]
Since \( 7 \) is between \( 4 \) and \( 9 \), we can estimate \( \sqrt{7} \) to be between \( 2 \) and \( 3 \). A more precise approximation gives:
\[
\sqrt{7} \approx 2.64575
\]
Now, we can calculate \( 4\sqrt{7} \):
\[
4\sqrt{7} \approx 4 \times 2.64575 \approx 10.583
\]
Thus, we have:
\[
\sqrt{112} \approx 10.583
\]
Now we can find integers that fit the inequality \( \square < 10.583 < \square \). The integers that satisfy this are:
\[
10 < \sqrt{112} < 11
\]
Therefore, we can fill in the squares:
\[
10 < \sqrt{112} < 11
\]
So the final answer is:
\[
10 < \sqrt{112} < 11
\]
Quick Answer
\( 10 < \sqrt{112} < 11 \)
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