Question

what integer is between \\(\frac{2\pi}{3}\\) and \\(\sqrt{11}\\)?

Explanation:

Step1: Calculate \( \frac{2\pi}{3} \)

We know that \( \pi \approx 3.14 \), so \( \frac{2\pi}{3} \approx \frac{2\times3.14}{3} \approx \frac{6.28}{3} \approx 2.09 \).

Step2: Calculate \( \sqrt{11} \)

We know that \( 3^2 = 9 \) and \( 4^2 = 16 \), so \( \sqrt{9} = 3 \) and \( \sqrt{16} = 4 \). Since \( 9 < 11 < 16 \), then \( 3 < \sqrt{11} < 4 \). More precisely, \( \sqrt{11} \approx 3.3166 \).

Step3: Find the integer between them

We have \( \frac{2\pi}{3} \approx 2.09 \) and \( \sqrt{11} \approx 3.3166 \). The integers between \( 2.09 \) and \( 3.3166 \) is \( 3 \) (since \( 2 < 2.09 < 3 < 3.3166 < 4 \)).

Answer:

3