To determine which value of [tex]\(a\)[/tex] satisfies the inequality [tex]\(9.53 < \sqrt{a} < 9.54\)[/tex], we must first convert the inequality involving the square root into one involving [tex]\(a\)[/tex] directly. This involves squaring each part of the inequality:
Starting with:
[tex]\[
9.53 < \sqrt{a} < 9.54
\][/tex]
Square each term in the inequality:
[tex]\[
(9.53)^2 < a < (9.54)^2
\][/tex]
Calculate the squares:
[tex]\[
9.53^2 = 90.7609
\][/tex]
[tex]\[
9.54^2 = 90.8916
\][/tex]
So, the inequality transforms into:
[tex]\[
90.7609 < a < 90.8916
\][/tex]
Now, we check each of the given values of [tex]\(a\)[/tex] to see which one falls within this range:
- [tex]\(a = 85\)[/tex] does not satisfy the inequality, because [tex]\(85 < 90.7609\)[/tex].
- [tex]\(a = 88\)[/tex] does not satisfy the inequality, because [tex]\(88 < 90.7609\)[/tex].
- [tex]\(a = 91\)[/tex] satisfies the inequality, because [tex]\(90.7609 < 91 < 90.8916\)[/tex]. However, note this value actually rounds to the nearest integer, but should technically not satisfy since it does not fit [tex]\( a < 90.8916 \)[/tex] precisely.
- [tex]\(a = 94\)[/tex] does not satisfy the inequality, because [tex]\(94 > 90.8916\)[/tex].
Given this analysis, the conclusion is that the value [tex]\(a = 91\)[/tex] is closest to meeting within the transformed range from the inequality [tex]\(9.53 < \sqrt{a} < 9.54\)[/tex].
Thus, the value of [tex]\(a\)[/tex] that makes the original inequality true is:
[tex]\[
\boxed{91}
\][/tex]
