To determine which of the numbers \( \sqrt{10} \), \( \sqrt{8} \), \( \sqrt{12} \), and \( \sqrt{7} \) lies between \( \pi \) and \( \sqrt{11} \), we can first approximate \( \pi \) and \( \sqrt{11} \).
-
Approximate \( \pi \):
\( \pi \approx 3.14 \) -
Calculate \( \sqrt{11} \):
\( \sqrt{11} \approx 3.3166 \)
Now we know we are looking for a number that is between approximately \( 3.14 \) and \( 3.3166 \).
Next, we can calculate the approximate values of the square roots:
- \( \sqrt{10} \approx 3.162 \)
- \( \sqrt{8} \approx 2.828 \)
- \( \sqrt{12} \approx 3.464 \)
- \( \sqrt{7} \approx 2.646 \)
Now we can determine which values fall between \( 3.14 \) and \( 3.3166 \):
- \( \sqrt{10} \approx 3.162 \) (fits within the range \( (3.14, 3.3166) \))
- \( \sqrt{8} \approx 2.828 \) (not in the range)
- \( \sqrt{12} \approx 3.464 \) (not in the range)
- \( \sqrt{7} \approx 2.646 \) (not in the range)
Therefore, the only number that is between \( \pi \) and \( \sqrt{11} \) is:
\[ \sqrt{10} \]