To determine how many natural numbers lie between [tex]\(33^2\)[/tex] and [tex]\(34^2\)[/tex], let's go through the process step-by-step.
1. Calculate [tex]\(33^2\)[/tex] and [tex]\(34^2\)[/tex]:
[tex]\[
33^2 = 1089
\][/tex]
[tex]\[
34^2 = 1156
\][/tex]
2. Identify the range of numbers:
We are looking for natural numbers that are strictly between 1089 and 1156. This means we should count the natural numbers starting from 1090 up to 1155.
3. Determine the number of natural numbers in this range:
To count how many numbers lie between 1089 and 1156, we subtract 1089 from 1156 and then subtract 1 (because we exclude both the endpoints 1089 and 1156 themselves):
[tex]\[
\text{Number of natural numbers} = 1156 - 1089 - 1
\][/tex]
[tex]\[
1156 - 1089 = 67 \quad (\text{but this count includes both ends})
\][/tex]
[tex]\[
67 - 1 = 66
\][/tex]
Thus, the number of natural numbers that lie between [tex]\(33^2\)[/tex] and [tex]\(34^2\)[/tex] is [tex]\(66\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{66} \][/tex]
