To determine the value of [tex]\( a \)[/tex] that makes the inequality [tex]\( 9.53 < \sqrt{a} < 9.54 \)[/tex] true, we need to find the range of [tex]\( a \)[/tex] that satisfies this inequality.
1. First, square the lower bound of the inequality [tex]\( 9.53 \)[/tex]:
[tex]\[
9.53^2 = 90.7609
\][/tex]
2. Next, square the upper bound of the inequality [tex]\( 9.54 \)[/tex]:
[tex]\[
9.54^2 = 91.0116
\][/tex]
So, the inequality [tex]\( 9.53 < \sqrt{a} < 9.54 \)[/tex] translates to:
[tex]\[
90.7609 < a < 91.0116
\][/tex]
3. Now we need to identify which of the given choices falls within the range [tex]\( (90.7609, 91.0116) \)[/tex]. The choices provided are:
- 85
- 88
- 91
- 94
4. Let's analyze each option to see if it falls within our calculated range:
- 85 is not within the range [tex]\( (90.7609, 91.0116) \)[/tex]
- 88 is not within the range [tex]\( (90.7609, 91.0116) \)[/tex]
- 91 is within the range [tex]\( (90.7609, 91.0116) \)[/tex]
- 94 is not within the range [tex]\( (90.7609, 91.0116) \)[/tex]
Thus, the value of [tex]\( a \)[/tex] that makes the inequality [tex]\( 9.53 < \sqrt{a} < 9.54 \)[/tex] true is:
[tex]\[
\boxed{91}
\][/tex]
