To determine the two numbers between which [tex]\(\sqrt{2}\)[/tex] lies, let's evaluate each given option:
A. 1.2 and 1.3:
- The square of 1.2 is [tex]\(1.2 \times 1.2 = 1.44\)[/tex].
- The square of 1.3 is [tex]\(1.3 \times 1.3 = 1.69\)[/tex].
Here, [tex]\(\sqrt{2}\)[/tex] is approximately 1.414. Since 1.414 lies between 1.2 and 1.3, this option is incorrect.
B. 1.4 and 1.5:
- The square of 1.4 is [tex]\(1.4 \times 1.4 = 1.96\)[/tex].
- The square of 1.5 is [tex]\(1.5 \times 1.5 = 2.25\)[/tex].
Here, [tex]\(\sqrt{2}\)[/tex] is approximately 1.414, which lies directly between 1.4 and 1.5. This option is correct.
C. 1.3 and 1.4:
- The square of 1.3 is [tex]\(1.3 \times 1.3 = 1.69\)[/tex].
- The square of 1.4 is [tex]\(1.4 \times 1.4 = 1.96\)[/tex].
Since [tex]\(\sqrt{2}\)[/tex] is approximately 1.414, it lies between 1.3 and 1.4, making this option incorrect.
D. 1.5 and 1.6:
- The square of 1.5 is [tex]\(1.5 \times 1.5 = 2.25\)[/tex].
- The square of 1.6 is [tex]\(1.6 \times 1.6 = 2.56\)[/tex].
Here, [tex]\(\sqrt{2}\)[/tex] is approximately 1.414, which does not lie between 1.5 and 1.6, so this option is incorrect.
Therefore, the correct answer is:
B. 1.4 and 1.5
