To determine which value of [tex]\(a\)[/tex] satisfies the inequality [tex]\( 9.53 < \sqrt{a} < 9.54 \)[/tex], let’s break it down step by step.
1. Establish the boundaries of the inequality:
[tex]\[
9.53 < \sqrt{a} < 9.54
\][/tex]
2. Square both sides of the inequality to eliminate the square root:
[tex]\[
(9.53)^2 < a < (9.54)^2
\][/tex]
This transforms the inequality to:
[tex]\[
90.8209 < a < 91.0116
\][/tex]
3. Determine the possible values of [tex]\(a\)[/tex] given in the problem:
[tex]\[
85, 88, 91, 94
\][/tex]
4. Check each of these values to see if they lie within the range 90.8209 to 91.0116:
- For [tex]\( a = 85 \)[/tex]:
[tex]\[
85 \notin (90.8209, 91.0116)
\][/tex]
- For [tex]\( a = 88 \)[/tex]:
[tex]\[
88 \notin (90.8209, 91.0116)
\][/tex]
- For [tex]\( a = 91 \)[/tex]:
[tex]\[
91 \in (90.8209, 91.0116)
\][/tex]
- For [tex]\( a = 94 \)[/tex]:
[tex]\[
94 \notin (90.8209, 91.0116)
\][/tex]
5. Conclude which value makes the inequality true:
Hence, the value of [tex]\(a\)[/tex] that satisfies the inequality [tex]\( 9.53 < \sqrt{a} < 9.54 \)[/tex] is:
[tex]\[
\boxed{91}
\][/tex]
