Using rational approximations, which statement is true?

A. [tex]\sqrt{12}\ \textless \ \pi[/tex]
B. [tex]\sqrt{16}\ \textgreater \ 4[/tex]
C. [tex]\sqrt{12}\ \textgreater \ \pi[/tex]
D. [tex]\sqrt{16}\ \textless \ 4[/tex]



Answer :

To determine which statement is true, we need to approximate the values of [tex]\(\sqrt{12}\)[/tex], [tex]\(\pi\)[/tex], and [tex]\(\sqrt{16}\)[/tex], and then compare them.

Let's start with [tex]\(\sqrt{12}\)[/tex] and [tex]\(\sqrt{16}\)[/tex]:
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex]

Next, recall the value of [tex]\(\pi\)[/tex]:
- [tex]\(\pi \approx 3.1416\)[/tex]

Now, let's evaluate each of the given inequalities:

1. [tex]\(\sqrt{12} < \pi\)[/tex]
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex] is not less than [tex]\(\pi \approx 3.1416\)[/tex].
- This statement is False.

2. [tex]\(\sqrt{16} > 4\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex] is not greater than 4.
- This statement is False.

3. [tex]\(\sqrt{12} > \pi\)[/tex]
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex] is indeed greater than [tex]\(\pi \approx 3.1416\)[/tex].
- This statement is True.

4. [tex]\(\sqrt{16} < 4\)[/tex]
- [tex]\(\sqrt{16} = 4\)[/tex] is not less than 4.
- This statement is False.

After evaluating each of the statements, we find that the true statement is:
[tex]\[ \sqrt{12} > \pi \][/tex]