To determine which of the given options lies between \( \pi \) and \( \sqrt{11} \), we first need to know the approximate values of these two constants:
-
Value of \( \pi \):
- \( \pi \approx 3.14 \)
-
Value of \( \sqrt{11} \):
- \( \sqrt{11} \approx 3.32 \)
Now, we need to find the approximate values of the options given:
-
\( \sqrt{12} \):
- \( \sqrt{12} \approx 3.46 \)
-
\( \sqrt{7} \):
- \( \sqrt{7} \approx 2.65 \)
-
\( \sqrt{8} \):
- \( \sqrt{8} \approx 2.83 \)
-
\( \sqrt{10} \):
- \( \sqrt{10} \approx 3.16 \)
Now we can check which of these fall between \( \pi \approx 3.14 \) and \( \sqrt{11} \approx 3.32 \):
- \( \sqrt{12} \approx 3.46 \): This is greater than \( \sqrt{11} \) and therefore not between.
- \( \sqrt{7} \approx 2.65 \): This is less than \( \pi \) and therefore not between.
- \( \sqrt{8} \approx 2.83 \): This is also less than \( \pi \) and therefore not between.
- \( \sqrt{10} \approx 3.16 \): This is between \( \pi \approx 3.14 \) and \( \sqrt{11} \approx 3.32 \).
Thus, the option that lies between \( \pi \) and \( \sqrt{11} \) is:
\( \sqrt{10} \).