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What is the largest integer whose square root is an irrational number between 3 and 4?(1 point)

Find a rational number between 2 and 214%0D%0A2%0D%0A1%0D%0A4%0D%0A using the method of adding the numbers together and dividing by 2. Write the answer as a mixed number.(1 point)

Find a rational number with one decimal place that is between 17−−√%0D%0A17%0D%0A and 18−−√%0D%0A18%0D%0A.(1 point)

Which of the following is between π%0D%0A𝜋%0D%0A and 11−−√%0D%0A11%0D%0A ?(1 point)%0D%0AResponses%0D%0A%0D%0A8–√%0D%0A8%0D%0AStart Root 8 End Root%0D%0A%0D%0A12−−√%0D%0A12%0D%0AStart Root 12 End Root%0D%0A%0D%0A7–√%0D%0A7%0D%0AStart Root 7 End Root%0D%0A%0D%0A10−−√%0D%0A10

3 answers

Let's tackle each of your questions one by one:

  1. Largest integer whose square root is an irrational number between 3 and 4:

    • The numbers 3 and 4 have squares of \(3^2 = 9\) and \(4^2 = 16\).
    • The integers between 9 and 16 are 10, 11, 12, 13, 14, and 15.
    • The largest integer among these is 15. The square root of 15, \(\sqrt{15}\), is an irrational number and falls between 3 and 4.

  2. Finding a rational number between 2 and 2.14:

    • To find a rational number between these two values, we can add them together and divide by 2: \[ x = \frac{2 + 2.14}{2} = \frac{4.14}{2} = 2.07 \]
    • As a mixed number, \(2.07\) can be represented as 2\(\frac{7}{100}\) or simply 2.07 as there are no whole parts.

  3. Finding a rational number with one decimal place between \( \sqrt{17} \) and \( \sqrt{18} \):

    • First, we calculate approximate values:
      • \( \sqrt{17} \approx 4.123 \)
      • \( \sqrt{18} \approx 4.242 \)
    • A rational number with one decimal place between these two values could be:
      • 4.2 (since it's between 4.123 and 4.242).

  4. Identifying which value falls between \( \pi \) and \( \sqrt{11} \):

    • We know that \( \pi \approx 3.14 \) and \( \sqrt{11} \approx 3.316 \).
    • Now we can evaluate the responses:
      • \( \sqrt{8} \approx 2.828 \) (not between)
      • \( \sqrt{12} \approx 3.464 \) (not between)
      • \( \sqrt{7} \approx 2.646 \) (not between)
      • \( \sqrt{10} \approx 3.162 \) (between 3.14 and 3.316)
    • Thus, \( \sqrt{10} \) is the value that falls between \( \pi \) and \( \sqrt{11} \).

In summary:

  1. The largest integer is 15.
  2. A rational number between 2 and 2.14 is 2 \( \frac{7}{100} \) (or 2.07).
  3. A rational number between \( \sqrt{17} \) and \( \sqrt{18} \) is 4.2.
  4. The value between \( \pi \) and \( \sqrt{11} \) is \( \sqrt{10} \).
1. Which of the following is an irrational number between 5 and 7: 6,12,27,36
27 squared

2. What is the largest integer whose square root is an irrational number between 3 and 4?
15

3. Find a rational number between 2 and 2 1/4 using the method of adding the numbers together and dividing by 2. Write the answer as a mixed number.
2 1/8

4. Find a rational number with one decimal place that is between 17 and 18.
4.2

5. Which of the following is between π and 11
10 squared

Let's address your statements and questions point by point:

  1. Irrational number between 5 and 7:

    • Your question appears to ask for an irrational number among the options given. However, 6, 12, 27, and 36 are all integers (and thus rational).
    • To find an irrational number between 5 and 7, we can consider numbers like \(\sqrt{36} = 6\) or \(\sqrt{50} \approx 7.07\) or any number like \(5.5\sqrt{2} \approx 6.5\).
    • None of the given options are irrational numbers. If you meant 27 squared (729), it's certainly not between 5 and 7, as it is another integer.

  2. Largest integer whose square root is an irrational number between 3 and 4:

    • Your answer of 15 is correct. The square root of 15 is approximately 3.872, which is indeed an irrational number between 3 and 4.

  3. Finding a rational number between 2 and 2 1/4:

    • To find a rational number between 2 and \(2 \frac{1}{4}\), first convert \(2 \frac{1}{4}\) to an improper fraction or decimal:
      • \(2 \frac{1}{4} = 2.25\)
    • Now calculate: \[ x = \frac{2 + 2.25}{2} = \frac{4.25}{2} = 2.125 \]
    • As a mixed number, 2.125 can be expressed as \(2 \frac{1}{8}\), which corresponds with your answer.

  4. Finding a rational number with one decimal place between 17 and 18:

    • The numbers 17 and 18 are integers, so the rational number with one decimal place that lies between 17 and 18 would be 17.5.
    • Your answer of 4.2 doesn't seem to fit this context, so it appears there may have been an error in the presented context.

  5. Which of the following is between \( \pi \) (approximately 3.14) and 11:

    • If "10 squared" refers to \(10^2 = 100\), which is certainly not between \( \pi \) and 11.
    • If we are looking just for a number that is between \( \pi \) and 11 without additional context from a list, any number from this range such as 4 or 6 could work. However, none of these given terms seem relevant from the context provided.

To summarize:

  1. No irrational number is provided between 5 and 7 among your options.
  2. Correctly answered, it's 15.
  3. Correctly answered as \(2 \frac{1}{8}\).
  4. The correct rational number between 17 and 18 should be 17.5.
  5. "10 squared" does not fit between \( \pi \) and 11.

Please specify any options or context if you need further explanations or calculations for the last question!