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. (1 point)
To find a rational number between 2 and 2 1/4 using the method of adding the numbers together and dividing by 2, we must first find the sum of 2 and 2 1/4:
2 + 2 1/4 = 2 + 9/4 = 8/4 + 9/4 = 17/4.
Now, divide 17/4 by 2:
(17/4) / 2 = 17/4 ÷ 2 = 17/4 × 1/2 = 17/8.
Thus, a rational number between 2 and 2 1/4 using the given method is 17/8.
Writing 17/8 as a mixed number, we convert it to a mixed fraction:
17 ÷ 8 = 2 remainder 1.
Thus, 17/8 as a mixed number is 2 1/8.
Find a rational number with one decimal place that is between √17 and √18.
To find a rational number with one decimal place between √17 and √18, we can use the method of averaging.
First, find the average of √17 and √18:
(√17 + √18) / 2
Now find the square root of 17 and 18:
√17 ≈ 4.123
√18 ≈ 4.243
Taking the average:
(4.123 + 4.243) / 2 ≈ 4.183
Thus, a rational number with one decimal place between √17 and √18 is approximately 4.2.
Which of the following is between π and √11? (1 point)
Responses
√12
√8
√10
√7
To determine which of the given options is between π and √11, we can compare the values.
π is approximately 3.14, and √11 is approximately 3.32.
Let's examine each option:
- √12 is approximately 3.464. This is larger than both π and √11, so it is not between them.
- √8 is approximately 2.828. This is smaller than both π and √11, so it is not between them.
- √10 is approximately 3.162. This is larger than π, but smaller than √11, so it is between them.
- √7 is approximately 2.646. This is smaller than both π and √11, so it is not between them.
Therefore, the option that is between π and √11 is √10.
Show an example of an irrational number between 9 and 11. (1 point)
Responses
√20
√99
√10
√100
To determine an example of an irrational number between 9 and 11, we can evaluate the given options:
- √20 ≈ 4.472. This is larger than both 9 and 11, so it is not between them.
- √99 ≈ 9.949. This is larger than both 9 and 11, so it is not between them.
- √10 ≈ 3.162. This is smaller than both 9 and 11, so it is not between them.
- √100 = 10. This is larger than both 9 and 11, so it is not between them.
Since none of the given options satisfy the requirement of being between 9 and 11, we need to find a different irrational number within this range.
An example of an irrational number between 9 and 11 is π. Pi, approximately 3.14159, is irrational and falls between 9 and 11.
What is the least integer whose square root is an irrational number between 5 and 7? (1 point)
Responses
13
25
6
26
To determine the least integer whose square root is an irrational number between 5 and 7, we need to evaluate the square roots of the given options:
- √13 ≈ 3.606: This is larger than 3, so it is not between 5 and 7.
- √25 = 5: This is not an irrational number, so it is not between 5 and 7.
- √6 ≈ 2.449: This is smaller than 3, so it is not between 5 and 7.
- √26 ≈ 5.099: This is not an irrational number, so it is not between 5 and 7.
Since none of the given options satisfy the requirement of having an irrational square root between 5 and 7, we need to find a different answer.
The next integer to check is 16. √16 = 4, which is not between 5 and 7.
Next, we check 17. √17 ≈ 4.123, which is between 5 and 7.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 17.
i need it to be one of the answers i gave.
Apologies for the confusion. Among the options you provided, the least integer whose square root is an irrational number between 5 and 7 is 26.
The square root of 26 is approximately 5.099, which is greater than 5 and less than 7.
Show an example of a rational number that is between √10 and √11.
3.1
16/5
3.4
10/3
To determine a rational number that is between √10 and √11, we need to compare the given options with the square roots.
√10 ≈ 3.162
√11 ≈ 3.317
Let's evaluate each option:
- 3.1 is not a rational number, as it contains a decimal portion, but it is smaller than both √10 and √11.
- 16/5 = 3.2, which is smaller than √11, but larger than √10. Therefore, it falls between the two square roots.
- 3.4 is not a rational number since it contains a decimal portion, but it is larger than both √10 and √11.
- 10/3 ≈ 3.333, which is larger than √10, but smaller than √11. Thus, it is between the two square roots.
Therefore, the rational number that is between √10 and √11 is 16/5.
Which irrational number is between √14 and √18?
√19
√15
√13
√10
To determine which irrational number is between √14 and √18, we need to compare the given options with the square roots:
√14 ≈ 3.742
√18 ≈ 4.243
Let's evaluate each option:
- √19 ≈ 4.359. This is larger than √18, so it is not between √14 and √18.
- √15 ≈ 3.873. This is larger than √14, but smaller than √18, so it falls between the two square roots.
- √13 ≈ 3.605. This is smaller than √14, so it is not between √14 and √18.
- √10 ≈ 3.162. This is smaller than √14, so it is not between √14 and √18.
Therefore, the irrational number that is between √14 and √18 is √15.
How many irrational numbers are there between √11 and √17?
twenty-eight numbers
four numbers
an infinite number
six numbers
To determine the number of irrational numbers between √11 and √17, we need to consider the nature of irrational numbers.
Irrational numbers are infinite and uncountable, meaning there are infinitely many and they cannot be listed or counted exhaustively. Thus, the correct answer is that there is an infinite number of irrational numbers between √11 and √17.