Answer :
Let's solve the problem of identifying the locations on the number line for the values [tex]\(\sqrt{12}\)[/tex], [tex]\(\sqrt{15}\)[/tex], and [tex]\(\frac{22}{9}\)[/tex], and match those values to the options given for points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
### Step 1: Calculate the Values
- [tex]\(\sqrt{12}\)[/tex]: The square root of 12 is approximately 3.4641.
- [tex]\(\sqrt{15}\)[/tex]: The square root of 15 is approximately 3.8730.
- [tex]\(\frac{22}{9}\)[/tex]: The fraction [tex]\(\frac{22}{9}\)[/tex] is approximately 2.4444.
### Step 2: Identify the Relative Positions on the Number Line
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex]
- [tex]\(\sqrt{15} \approx 3.8730\)[/tex]
- [tex]\(\frac{22}{9} \approx 2.4444\)[/tex]
Clearly, comparing these values:
- [tex]\(\frac{22}{9} \approx 2.4444\)[/tex] is the smallest.
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex] is in the middle.
- [tex]\(\sqrt{15} \approx 3.8730\)[/tex] is the largest.
### Step 3: Match the Points with the Values
Review the four given options:
Option 1:
- Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]).
This matches the values exactly as calculated, so this is a correct arrangement.
Option 2:
- Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]).
In this option, point [tex]\(A\)[/tex] is the smallest value [tex]\(\frac{22}{9}\)[/tex], which also follows the relative positions well.
Option 3:
- Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]).
This arrangement does not respect the correct positions because point [tex]\(B\)[/tex] should be the smallest, but here it is placed in the middle.
Option 4:
- Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]).
This option also correctly follows the order on the number line.
### Conclusion:
The correct points arrangement based on the values and the relative positions on the number line are:
- Option 1: Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex].
- Option 2: Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{12}\)[/tex].
- Option 4: Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{12}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex].
However, options 3 does not follow the correct positional order based on the calculated values.
### Step 1: Calculate the Values
- [tex]\(\sqrt{12}\)[/tex]: The square root of 12 is approximately 3.4641.
- [tex]\(\sqrt{15}\)[/tex]: The square root of 15 is approximately 3.8730.
- [tex]\(\frac{22}{9}\)[/tex]: The fraction [tex]\(\frac{22}{9}\)[/tex] is approximately 2.4444.
### Step 2: Identify the Relative Positions on the Number Line
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex]
- [tex]\(\sqrt{15} \approx 3.8730\)[/tex]
- [tex]\(\frac{22}{9} \approx 2.4444\)[/tex]
Clearly, comparing these values:
- [tex]\(\frac{22}{9} \approx 2.4444\)[/tex] is the smallest.
- [tex]\(\sqrt{12} \approx 3.4641\)[/tex] is in the middle.
- [tex]\(\sqrt{15} \approx 3.8730\)[/tex] is the largest.
### Step 3: Match the Points with the Values
Review the four given options:
Option 1:
- Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]).
This matches the values exactly as calculated, so this is a correct arrangement.
Option 2:
- Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]).
In this option, point [tex]\(A\)[/tex] is the smallest value [tex]\(\frac{22}{9}\)[/tex], which also follows the relative positions well.
Option 3:
- Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]).
This arrangement does not respect the correct positions because point [tex]\(B\)[/tex] should be the smallest, but here it is placed in the middle.
Option 4:
- Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex] ([tex]\(\approx 2.4444\)[/tex]),
- Point [tex]\(B\)[/tex] is [tex]\(\sqrt{12}\)[/tex] ([tex]\(\approx 3.4641\)[/tex]),
- Point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex] ([tex]\(\approx 3.8730\)[/tex]).
This option also correctly follows the order on the number line.
### Conclusion:
The correct points arrangement based on the values and the relative positions on the number line are:
- Option 1: Point [tex]\(A\)[/tex] is [tex]\(\sqrt{12}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\frac{22}{9}\)[/tex].
- Option 2: Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{15}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{12}\)[/tex].
- Option 4: Point [tex]\(A\)[/tex] is [tex]\(\frac{22}{9}\)[/tex], point [tex]\(B\)[/tex] is [tex]\(\sqrt{12}\)[/tex], and point [tex]\(C\)[/tex] is [tex]\(\sqrt{15}\)[/tex].
However, options 3 does not follow the correct positional order based on the calculated values.
