Which set of numbers is always rational?(1 point)
Responses
decimals
decimals
integers
integers
positive numbers
positive numbers
fractions
31 answers
fractions
The height of an equilateral triangle equals 3√2 times the side length. Using 74 for the square root of 3, what is the approximate height in meters of an equilateral triangle with a side length of 8 meters?(1 point)
The height of an equilateral triangle is given by the formula:
height = √3/2 * side length
Given that the side length is 8 meters, we can substitute this value into the formula:
height = √3/2 * 8
Next, we substitute the value of 74 for the square root of 3:
height = 74/2 * 8
Simplifying the fraction:
height = 37 * 8
Multiplying:
height = 296
Therefore, the height of the equilateral triangle is approximately 296 meters.
height = √3/2 * side length
Given that the side length is 8 meters, we can substitute this value into the formula:
height = √3/2 * 8
Next, we substitute the value of 74 for the square root of 3:
height = 74/2 * 8
Simplifying the fraction:
height = 37 * 8
Multiplying:
height = 296
Therefore, the height of the equilateral triangle is approximately 296 meters.
The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)
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The diagonal of a square is given by the formula:
diagonal = √2 * side length
Given that the side length is 700 feet, we can substitute this value into the formula:
diagonal = 1.4 * 700
Multiplying:
diagonal = 980
Therefore, the approximate length in feet from one corner to an opposite corner of the square park is 980 feet.
diagonal = √2 * side length
Given that the side length is 700 feet, we can substitute this value into the formula:
diagonal = 1.4 * 700
Multiplying:
diagonal = 980
Therefore, the approximate length in feet from one corner to an opposite corner of the square park is 980 feet.
show an example of an irrational number between 9 and 11.
One example of an irrational number between 9 and 11 is √10. When approximated, √10 ≈ 3.162.
Therefore, √10 is an irrational number between 9 and 11.
Therefore, √10 is an irrational number between 9 and 11.
What is the least integer whose square root is an irrational number between 5 and 7?
To find the least integer whose square root is an irrational number between 5 and 7, we need to square each of the given numbers and find the smallest perfect square that lies between those values.
Squaring 5, we get 25.
Squaring 6, we get 36.
Squaring 7, we get 49.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 6.
Squaring 5, we get 25.
Squaring 6, we get 36.
Squaring 7, we get 49.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 6.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
On a number line, we can identify the whole number and rational number to the nearest half that 40√ lies between.
To find the whole number and rational number, we calculate the square root of 40 and round to the nearest half.
The square root of 40 is approximately 6.325. Rounded to the nearest half, it is 6.5. Therefore, 40√ lies between 6 and 7 on the number line.
Thus, the whole number is 6, and the rational number to the nearest half is 6.5.
To find the whole number and rational number, we calculate the square root of 40 and round to the nearest half.
The square root of 40 is approximately 6.325. Rounded to the nearest half, it is 6.5. Therefore, 40√ lies between 6 and 7 on the number line.
Thus, the whole number is 6, and the rational number to the nearest half is 6.5.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?(1 point)
Responses
between 8 and 8.5
between 8 and 8.5
between 8.5 and 9
between 8.5 and 9
between 7.5 and 8
between 7.5 and 8
between 7 and 7.5
Responses
between 8 and 8.5
between 8 and 8.5
between 8.5 and 9
between 8.5 and 9
between 7.5 and 8
between 7.5 and 8
between 7 and 7.5
To determine the whole number and rational number to the nearest half that 58√ lies between, we need to calculate the square root of 58 and round to the nearest half.
The square root of 58 is approximately 7.615. Rounded to the nearest half, it is 7.5.
Therefore, 58√ lies between 7 and 7.5 on the number line.
Thus, the whole number is 7, and the rational number to the nearest half is 7.5.
The square root of 58 is approximately 7.615. Rounded to the nearest half, it is 7.5.
Therefore, 58√ lies between 7 and 7.5 on the number line.
Thus, the whole number is 7, and the rational number to the nearest half is 7.5.
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
We can use rational approximations to compare 81√ and √8.
√81 is equal to 9, which is a whole number.
To approximate √8, we can use the fact that √9 is 3 and √16 is 4. Therefore, √8 lies between 3 and 4.
Since 9 is greater than 3, we can conclude that 81√ is greater than √8.
Therefore, the inequality statement is:
81√ > √8
√81 is equal to 9, which is a whole number.
To approximate √8, we can use the fact that √9 is 3 and √16 is 4. Therefore, √8 lies between 3 and 4.
Since 9 is greater than 3, we can conclude that 81√ is greater than √8.
Therefore, the inequality statement is:
81√ > √8
Using rational approximations, what is an inequality statement comparing 81−−√ and –√88? Enter <, >, or =.(1 point)
To compare 81√ and –√88 using rational approximations, we need to find the nearest rational approximations for each square root.
The square root of 81 is 9, which is a whole number.
For –√88, we can approximate it by considering √81, which is 9, and √100, which is 10. Therefore, –√88 lies between –9 and –10.
Since 9 is greater than –9, we can conclude that 81√ is greater than –√88.
Therefore, the inequality statement is:
81√ > –√88
The square root of 81 is 9, which is a whole number.
For –√88, we can approximate it by considering √81, which is 9, and √100, which is 10. Therefore, –√88 lies between –9 and –10.
Since 9 is greater than –9, we can conclude that 81√ is greater than –√88.
Therefore, the inequality statement is:
81√ > –√88
(9×1,000)+(3×100)+(7×10)+(6×1)
left parenthesis 9 times 1,000 right parenthesis plus left parenthesis 3 times 100 right parenthesis plus left parenthesis 7 times 10 right parenthesis plus left parenthesis 6 times 1 right parenthesis
(7×100)+(3×10)+(9×1)
left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(9×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(0×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 0 times 1 right parenthesis
left parenthesis 9 times 1,000 right parenthesis plus left parenthesis 3 times 100 right parenthesis plus left parenthesis 7 times 10 right parenthesis plus left parenthesis 6 times 1 right parenthesis
(7×100)+(3×10)+(9×1)
left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(9×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 9 times 1 right parenthesis
(6×1,000)+(7×100)+(3×10)+(0×1)
left parenthesis 6 times 1,000 right parenthesis plus left parenthesis 7 times 100 right parenthesis plus left parenthesis 3 times 10 right parenthesis plus left parenthesis 0 times 1 right parenthesis
(9×1,000)+(3×100)+(7×10)+(6×1) simplifies to:
9,000 + 300 + 70 + 6 = 9,376
(7×100)+(3×10)+(9×1) simplifies to:
700 + 30 + 9 = 739
(6×1,000)+(7×100)+(3×10)+(9×1) simplifies to:
6,000 + 700 + 30 + 9 = 6,739
(6×1,000)+(7×100)+(3×10)+(0×1) simplifies to:
6,000 + 700 + 30 + 0 = 6,730
9,000 + 300 + 70 + 6 = 9,376
(7×100)+(3×10)+(9×1) simplifies to:
700 + 30 + 9 = 739
(6×1,000)+(7×100)+(3×10)+(9×1) simplifies to:
6,000 + 700 + 30 + 9 = 6,739
(6×1,000)+(7×100)+(3×10)+(0×1) simplifies to:
6,000 + 700 + 30 + 0 = 6,730
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.
To express 811 as a repeating decimal, we can start by dividing 811 by the denominator of the fraction for the repeating decimal. In this case, the denominator is 9 because the repeating part has one digit.
Here's the division:
```
_____
9 | 811
- 9
-----
71
- 63
-----
800
- 720
-----
800
- 720
-----
80
```
The remainder becomes 80. To express this as a repeating decimal, we add a zero to the remainder and divide by 9 again:
```
_____
9 | 800
- 9
-----
100
- 99
-----
10
```
The remainder now becomes 10. We can continue this process by adding zeroes and dividing by 9 until we reach a remainder of zero:
```
_____
9 | 100
- 99
-----
1
```
Since the remainder is 1, the repeating cycle is determined to be 1.
Therefore, 811 expressed as a repeating decimal is approximately 90.1̅, where the bar is placed over the digit 1 to mark the repeating cycle.
Here's the division:
```
_____
9 | 811
- 9
-----
71
- 63
-----
800
- 720
-----
800
- 720
-----
80
```
The remainder becomes 80. To express this as a repeating decimal, we add a zero to the remainder and divide by 9 again:
```
_____
9 | 800
- 9
-----
100
- 99
-----
10
```
The remainder now becomes 10. We can continue this process by adding zeroes and dividing by 9 until we reach a remainder of zero:
```
_____
9 | 100
- 99
-----
1
```
Since the remainder is 1, the repeating cycle is determined to be 1.
Therefore, 811 expressed as a repeating decimal is approximately 90.1̅, where the bar is placed over the digit 1 to mark the repeating cycle.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert 0.72̅ to a rational number in simplest form, we can follow these steps:
Let x = 0.72̅
Step 1: Multiply both sides of the equation by a power of 10 to eliminate the repeating decimal. In this case, we multiply by 100:
100x = 72.72̅
Step 2: Subtract the original equation (x) from the equation obtained in step 1 (100x):
100x - x = 72.72̅ - 0.72̅
99x = 72
Step 3: Divide both sides of the equation by 99 to solve for x:
x = 72/99
To simplify further, we can divide both the numerator and the denominator by their greatest common divisor, which is 9:
x = (8 * 9) / (11 * 9)
x = 8/11
Therefore, 0.72̅ can be expressed as the rational number 8/11 in simplest form.
Let x = 0.72̅
Step 1: Multiply both sides of the equation by a power of 10 to eliminate the repeating decimal. In this case, we multiply by 100:
100x = 72.72̅
Step 2: Subtract the original equation (x) from the equation obtained in step 1 (100x):
100x - x = 72.72̅ - 0.72̅
99x = 72
Step 3: Divide both sides of the equation by 99 to solve for x:
x = 72/99
To simplify further, we can divide both the numerator and the denominator by their greatest common divisor, which is 9:
x = (8 * 9) / (11 * 9)
x = 8/11
Therefore, 0.72̅ can be expressed as the rational number 8/11 in simplest form.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
525
5 Start Fraction 2 over 5 end fraction
26−−√
start root 26 end root
513
5 Start Fraction 1 over 3 end fraction
517
Responses
525
5 Start Fraction 2 over 5 end fraction
26−−√
start root 26 end root
513
5 Start Fraction 1 over 3 end fraction
517
To find a rational number between 5 and 514, we can look for a number that is not only between the two values but can also be expressed as a fraction.
Out of the given options, the rational number between 5 and 514 is 5 2/5. This can be seen as a fraction and is between the given values.
Therefore, the correct option is: 5 Start Fraction 2 over 5 end fraction.
Out of the given options, the rational number between 5 and 514 is 5 2/5. This can be seen as a fraction and is between the given values.
Therefore, the correct option is: 5 Start Fraction 2 over 5 end fraction.
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
To find a rational number with a denominator of 7 between 7√ and 8√, we can consider the square root of 49 and the square root of 64.
The square root of 49 is 7, and the square root of 64 is 8. Therefore, we need to find a rational number with a denominator of 7 between 7 and 8.
One such rational number is 15/7.
Therefore, the rational number with a denominator of 7 that is between 7√ and 8√ is 15/7.
The square root of 49 is 7, and the square root of 64 is 8. Therefore, we need to find a rational number with a denominator of 7 between 7 and 8.
One such rational number is 15/7.
Therefore, the rational number with a denominator of 7 that is between 7√ and 8√ is 15/7.
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.(4 points)
Rational numbers and whole numbers are related, but they are not the same.
1. Not all rational numbers are whole numbers: Rational numbers include fractions and decimals that can be expressed as a ratio of two integers. Whole numbers, on the other hand, are non-negative integers without any fractional parts. For example, the rational number 3/2 is not a whole number because it has a fractional part.
2. All whole numbers are rational numbers: Whole numbers are inclusive of all non-negative integers (e.g., 0, 1, 2, 3, ...), and integers can be expressed as the ratio of an integer divided by 1. For instance, the whole number 5 can be expressed as 5/1, making it a rational number.
To summarize, all whole numbers are rational numbers since they can be expressed as a fraction with a denominator of 1. However, not all rational numbers are whole numbers since they can possess fractional parts.
In conclusion, the relationship is that all whole numbers are a subset of rational numbers, but not all rational numbers are whole numbers.
1. Not all rational numbers are whole numbers: Rational numbers include fractions and decimals that can be expressed as a ratio of two integers. Whole numbers, on the other hand, are non-negative integers without any fractional parts. For example, the rational number 3/2 is not a whole number because it has a fractional part.
2. All whole numbers are rational numbers: Whole numbers are inclusive of all non-negative integers (e.g., 0, 1, 2, 3, ...), and integers can be expressed as the ratio of an integer divided by 1. For instance, the whole number 5 can be expressed as 5/1, making it a rational number.
To summarize, all whole numbers are rational numbers since they can be expressed as a fraction with a denominator of 1. However, not all rational numbers are whole numbers since they can possess fractional parts.
In conclusion, the relationship is that all whole numbers are a subset of rational numbers, but not all rational numbers are whole numbers.
1 answer