Which set of numbers is always rational?(1 point)
Responses
positive numbers
positive numbers
decimals
decimals
integers
integers
fractions
23 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?
Given that the height of an equilateral triangle is equal to 3√2 times the side length, we can calculate the height by multiplying 3√2 by the side length.
First, we need to approximate the value of 3√2 using the given approximation of the square root of 3: 74. We can write 3√2 as 3 * √2.
Multiplying 3 by 74 gives us 222. Therefore, the approximate value of 3 * √2 is 222.
Now, we can find the height by multiplying the side length of the equilateral triangle (8 meters) by 222: 8 * 222 = 1776.
Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 1776 meters.
First, we need to approximate the value of 3√2 using the given approximation of the square root of 3: 74. We can write 3√2 as 3 * √2.
Multiplying 3 by 74 gives us 222. Therefore, the approximate value of 3 * √2 is 222.
Now, we can find the height by multiplying the side length of the equilateral triangle (8 meters) by 222: 8 * 222 = 1776.
Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 1776 meters.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
40−−√ is between
and
.
40−−√ is between
and
.
To find the whole number and rational number to the nearest half that 40−−√ lies between, we can approximate the value of 40−−√.
We know that the square root of 40 is between 6 and 7 since 6^2 = 36 and 7^2 = 49.
To get an estimation of 40−−√, we can take the average of 6 and 7:
(6 + 7) / 2 = 6.5
Therefore, 40−−√ is approximately 6.5.
The whole number to the nearest half that 6.5 is between is 6, and the rational number to the nearest half is 6.5 itself.
So, 40−−√ is between 6 and 6.5.
We know that the square root of 40 is between 6 and 7 since 6^2 = 36 and 7^2 = 49.
To get an estimation of 40−−√, we can take the average of 6 and 7:
(6 + 7) / 2 = 6.5
Therefore, 40−−√ is approximately 6.5.
The whole number to the nearest half that 6.5 is between is 6, and the rational number to the nearest half is 6.5 itself.
So, 40−−√ is between 6 and 6.5.
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
The inequality is: 81−−√
8–√8.
The inequality is: 81−−√
8–√8.
To compare 81−−√ and 8–√8 using rational approximations, we can find their approximate values.
For 81−−√, we can approximate the square root of 81 as 9.
For 8–√8, we can approximate the square root of 8 as 2.83 (rounded to two decimal places).
Therefore, the inequality statement comparing 81−−√ and 8–√8 is:
81−−√ > 8–√8.
For 81−−√, we can approximate the square root of 81 as 9.
For 8–√8, we can approximate the square root of 8 as 2.83 (rounded to two decimal places).
Therefore, the inequality statement comparing 81−−√ and 8–√8 is:
81−−√ > 8–√8.
Monique measured the widths of several stamps in her collection of world stamps. The measurements are 158in., 11116in., 134in., and 112in. What is the width of the widest stamp?
To find the width of the widest stamp, we need to compare the given measurements and determine the largest one.
The measurements given are:
158in., 11116in., 134in., and 112in.
To compare these measurements, we can list them in ascending order:
112in., 134in., 158in., 11116in.
The largest measurement is 11116in.
Therefore, the width of the widest stamp is 11116in.
The measurements given are:
158in., 11116in., 134in., and 112in.
To compare these measurements, we can list them in ascending order:
112in., 134in., 158in., 11116in.
The largest measurement is 11116in.
Therefore, the width of the widest stamp is 11116in.
What is the circumference of a circle with a diameter of 4.5 miles? Estimate the circumference to the hundredths place using 3.14 for π
.(1 point)
Responses
63.59 miles
63.59 miles
14.2 miles
14.2 miles
14.13 miles
14.13 miles
7.07 miles
.(1 point)
Responses
63.59 miles
63.59 miles
14.2 miles
14.2 miles
14.13 miles
14.13 miles
7.07 miles
The circumference of a circle can be calculated using the formula: circumference = π * diameter.
Given that the diameter of the circle is 4.5 miles, we can calculate the circumference using the approximation of π as 3.14.
circumference ≈ 3.14 * 4.5 = 14.13
Therefore, the circumference of a circle with a diameter of 4.5 miles is approximately 14.13 miles.
Given that the diameter of the circle is 4.5 miles, we can calculate the circumference using the approximation of π as 3.14.
circumference ≈ 3.14 * 4.5 = 14.13
Therefore, the circumference of a circle with a diameter of 4.5 miles is approximately 14.13 miles.
Write 6,739 in expanded form.(1 point)
Responses
(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
(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
(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)
Responses
(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
(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
(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)
(6×1,000)+(7×100)+(3×10)+(9×1)
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.
To express 811 as a repeating decimal, we divide 811 by 990 (since the repeating cycle of digits is 990):
811 ÷ 990 = 0.818181...
The repeating cycle of digits in the decimal representation of 811 is "81". Therefore, we can express 811 as a repeating decimal as 0.81¯.
811 ÷ 990 = 0.818181...
The repeating cycle of digits in the decimal representation of 811 is "81". Therefore, we can express 811 as a repeating decimal as 0.81¯.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can set x = 0.72¯¯¯¯¯ and manipulate the equation to eliminate the repeating part.
Let's multiply both sides of the equation x = 0.72¯¯¯¯¯ by 1000 to clear the repeating digits:
1000x = 720.72¯¯¯¯¯
Next, we subtract the original equation from the modified equation to eliminate the repeating part:
1000x - x = 720.72¯¯¯¯¯ - 0.72¯¯¯¯¯
999x = 720
Now, we can solve for x by dividing both sides of the equation by 999:
x = 720 / 999
The fraction 720/999 is already in its simplest form, as the numerator and denominator share no common factors other than 1.
Therefore, the rational number in simplest form equivalent to the repeating decimal 0.72¯¯¯¯¯ is 720/999.
Let's multiply both sides of the equation x = 0.72¯¯¯¯¯ by 1000 to clear the repeating digits:
1000x = 720.72¯¯¯¯¯
Next, we subtract the original equation from the modified equation to eliminate the repeating part:
1000x - x = 720.72¯¯¯¯¯ - 0.72¯¯¯¯¯
999x = 720
Now, we can solve for x by dividing both sides of the equation by 999:
x = 720 / 999
The fraction 720/999 is already in its simplest form, as the numerator and denominator share no common factors other than 1.
Therefore, the rational number in simplest form equivalent to the repeating decimal 0.72¯¯¯¯¯ is 720/999.
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
One rational number between 5 and 514 is 5 Start Fraction 2 over 5 end fraction.
Therefore, the correct response is: 5 Start Fraction 2 over 5 end fraction.
Therefore, the correct response 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.
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we can first simplify the given square roots:
7–√ = (√7) / 7
8–√ = (√8) / 7
To find a rational number between these two values, we need to find the average:
[ (√7) / 7 + (√8) / 7 ] / 2
Now, let's simplify the numerator:
√7 + √8
To rationalize the denominator, we multiply both the numerator and denominator by (√7 - √8):
[ (√7 + √8) * (√7 - √8) ] / (7 * 7)
Simplifying further, using the difference of squares formula (√a + √b)(√a - √b) = a - b:
(7 - 8) / (7 * 7)
(-1) / 49
Therefore, a rational number with a denominator of 7 between 7–√ and 8–√ is -1/49.
7–√ = (√7) / 7
8–√ = (√8) / 7
To find a rational number between these two values, we need to find the average:
[ (√7) / 7 + (√8) / 7 ] / 2
Now, let's simplify the numerator:
√7 + √8
To rationalize the denominator, we multiply both the numerator and denominator by (√7 - √8):
[ (√7 + √8) * (√7 - √8) ] / (7 * 7)
Simplifying further, using the difference of squares formula (√a + √b)(√a - √b) = a - b:
(7 - 8) / (7 * 7)
(-1) / 49
Therefore, a rational number with a denominator of 7 between 7–√ and 8–√ is -1/49.
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.
Rational numbers and whole numbers are related, but they are not the same.
All whole numbers are rational numbers because they can be expressed as a ratio of two integers, where the denominator is 1. For example, 3 can be expressed as 3/1, which is a ratio of two integers, making it a rational number.
However, not all rational numbers are whole numbers. Rational numbers include fractions, decimals, and integers. Fractions and decimals that are not whole numbers are also rational numbers. For example, 1/2 and 0.75 are rational numbers but not whole numbers.
To summarize:
- All whole numbers are rational numbers.
- Not all rational numbers are whole numbers.
Examples:
Whole number: 7
Rational number (fraction): 3/5
Rational number (decimal): 1.2 (not a whole number)
All whole numbers are rational numbers because they can be expressed as a ratio of two integers, where the denominator is 1. For example, 3 can be expressed as 3/1, which is a ratio of two integers, making it a rational number.
However, not all rational numbers are whole numbers. Rational numbers include fractions, decimals, and integers. Fractions and decimals that are not whole numbers are also rational numbers. For example, 1/2 and 0.75 are rational numbers but not whole numbers.
To summarize:
- All whole numbers are rational numbers.
- Not all rational numbers are whole numbers.
Examples:
Whole number: 7
Rational number (fraction): 3/5
Rational number (decimal): 1.2 (not a whole number)