To determine which of the given options is an irrational number \( d \) that satisfies \( 6 < d < 7 \), we can evaluate each option:
A. \( \sqrt{6.2} \)
Calculating \( \sqrt{6.2} \):
- Since \( 6.2 \) is between \( 6 \) and \( 7 \), its square root will also be between \( \sqrt{6} \) and \( \sqrt{7} \).
- Since \( \sqrt{6} \approx 2.45 \) and \( \sqrt{7} \approx 2.65 \), it follows that \( \sqrt{6.2} \approx 2.49 \), which is much less than \( 7 \). Thus, \( \sqrt{6.2} < 6.5 \).
Not a valid option.
B. \( 2\pi \)
Calculating \( 2\pi \):
- \( 2\pi \approx 6.283 \), which is indeed between \( 6 \) and \( 7 \).
- This number is also irrational.
Valid option.
C. \( \sqrt{13} \)
Calculating \( \sqrt{13} \):
- \( \sqrt{13} \approx 3.605 \), which is much less than \( 6.5 \).
Not a valid option.
D. \( 6\pi \)
Calculating \( 6\pi \):
- \( 6\pi \approx 18.85 \), which is much larger than \( 7 \).
Not a valid option.
Considering all of the evaluations, the only option that fits the criteria \( 6 < d < 7 \) and is an irrational number is:
B. \( 2\pi \)