To determine which answer choice lists the given set of numbers in correct ascending order, we need to first understand their approximate numerical values.
Let's start by converting each expression to its numerical equivalent:
1. [tex]\(1 \frac{3}{4}\)[/tex] is a mixed number that can be converted to an improper fraction:
[tex]\[
1 + \frac{3}{4} = 1.75
\][/tex]
2. [tex]\(\sqrt{\pi}\)[/tex] represents the square root of [tex]\(\pi\)[/tex]. Using the approximate value of [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
\sqrt{3.14159} \approx 1.772
\][/tex]
3. The number 1.71 is already in decimal form.
4. [tex]\(\frac{16}{9}\)[/tex] is a fraction:
[tex]\[
\frac{16}{9} \approx 1.777
\][/tex]
Now, we list these values in simplified numeric form for comparison:
- [tex]\(1 \frac{3}{4} = 1.75\)[/tex]
- [tex]\(\sqrt{\pi} \approx 1.772\)[/tex]
- [tex]\(1.71\)[/tex]
- [tex]\(\frac{16}{9} \approx 1.777\)[/tex]
Next, we will order these numbers from least to greatest:
- [tex]\(1.71\)[/tex]
- [tex]\(1.75\)[/tex]
- [tex]\(1.772\)[/tex]
- [tex]\(1.777\)[/tex]
Now we will check each answer choice to see which correctly lists the numbers in ascending order:
Choice 1: [tex]\(1 \frac{3}{4}, \sqrt{\pi}, 1.71, \frac{16}{9}\)[/tex]
- This sequence is: 1.75, 1.772, 1.71, 1.777
- Clearly, the order is incorrect.
Choice 2: [tex]\(\sqrt{\pi}, 1 \frac{3}{4}, \frac{16}{9}, 1.71\)[/tex]
- This sequence is: 1.772, 1.75, 1.777, 1.71
- The order is incorrect.
Choice 3: [tex]\(1.71, 1 \frac{3}{4}, \sqrt{\pi}, \frac{16}{9}\)[/tex]
- This sequence is: 1.71, 1.75, 1.772, 1.777
- The order is correct.
Choice 4: [tex]\(\frac{16}{9} \cdot 1.71, \sqrt{\pi}, 1 \frac{3}{4}\)[/tex]
- This is an incomplete option and thus can't be valid.
Thus, the correct answer choice that represents these numbers in ascending order from least to greatest is:
Choice 3: [tex]\(1.71, 1 \frac{3}{4}, \sqrt{\pi}, \frac{16}{9}\)[/tex]
