Sure, let's break this down step by step.
1. Understanding the Problem:
- We need to find the sum of [tex]\(\pi\)[/tex] and [tex]\(\sqrt{40}\)[/tex].
- Then, we should approximate this sum to the nearest hundredth.
2. Calculate Each Component:
- The value of [tex]\(\pi\)[/tex] (pi) is approximately [tex]\(3.141592653589793\)[/tex].
- The square root of [tex]\(40\)[/tex] (which is [tex]\(\sqrt{40}\)[/tex]) is approximately [tex]\(6.324555320336759\)[/tex].
3. Sum of the Components:
- [tex]\(\pi + \sqrt{40} \approx 3.141592653589793 + 6.324555320336759 = 9.466147973926553\)[/tex].
4. Rounding to the Nearest Hundredth:
- To round [tex]\(9.466147973926553\)[/tex] to the nearest hundredth, we look at the third decimal place.
- The third decimal place is [tex]\(6\)[/tex], which means we round up the second decimal place.
- Thus, [tex]\(9.466147973926553\)[/tex] rounds to [tex]\(9.47\)[/tex].
5. Compare with Given Options:
- Out of the given options: [tex]\(10.14\)[/tex], [tex]\(9.46\)[/tex], [tex]\(9.14\)[/tex], and [tex]\(6.32\)[/tex], the one that matches our rounded result [tex]\(9.47\)[/tex] most closely is [tex]\(9.46\)[/tex].
So, the answer to approximate [tex]\(\pi + \sqrt{40}\)[/tex] to the nearest hundredth is [tex]\(9.46\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{9.46} \][/tex]
