Asked by Kozylovesyou

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).
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The height of an equilateral triangle equals 3√2 times the side length. Using 7/4 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)
Answered by Kozylovesyou
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 poi
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Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.
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Monique measured the widths of several stamps in her collection of world stamps. The measurements are 1 5/8in., 1 11/16in., 1 1 1 1 3/4in., and 1 1/2in. What is the width of the widest stamp?(1 point)
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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
.
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Write 6,739 in expanded form.
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Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.
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0.72 in simplest form
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Convert 0.72¯¯¯¯¯ to a rational number in simplest form
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Which of the following is a rational number between 5 and 5 1/4?
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What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(
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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.
Answered by GPT 3.5
The height of an equilateral triangle equals 3√2 times the side length.

Given that the side length is 8 meters, we can substitute this value into the equation:

Height = 3√2 * Side length
Height = 3√2 * 8

Now, we just need to substitute the value of √2 with the given value of 74:

Height = 3 * (74) * 8
Height = 222 * 8
Height = 1776

Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 1776 meters.
Answered by GPT 3.5
Given that the square root of 3 is represented as 7/4, we can substitute this value into the equation:

Height = 3√2 * Side length
Height = 3 * (7/4) * 8

To simplify the equation, we can first multiply 3 and 8:

Height = 24 * (7/4)

Next, we can simplify the expression (7/4):

Height = (24 * 7) / 4
Height = 168 / 4
Height = 42

Therefore, the approximate height of an equilateral triangle with a side length of 8 meters, using 7/4 for the square root of 3, is 42 meters.
Answered by GPT 3.5
To determine which whole number and rational number to the nearest half the square root of 40 lies between, we can look at the number line.

By calculating the square root of 40, we find that it is approximately 6.32456.

On the number line, we can see that 6 is the whole number right before 6.32456, and 7 is the whole number right after it.

To find the nearest half, we can divide the interval between 6 and 7 into halves:

6 ------------ 6.5 ------------ 7

Since 6.32456 is closer to 6.5 than it is to 6, the square root of 40, rounded to the nearest half, is between 6 and 6.5.

Therefore, the whole number to the nearest half is 6, and the rational number to the nearest half is 6.5.
Answered by GPT 3.5
To compare 81−−√ and 8–√8 using rational approximations, we can find their decimal approximations.

The square root of 81 is exactly 9.
The square root of 8 is approximately 2.82843.

Comparing these decimal approximations, we can see that 9 is greater than 2.82843.

Therefore, the inequality statement comparing 81−−√ and 8–√8 is:

81−−√ > 8–√8
Answered by GPT 3.5
To find the width of the widest stamp among the given measurements, we need to compare the measurements and determine the largest one.

The given measurements are:
1 5/8in.
1 11/16in.
1 1 1 1 3/4in.
1 1/2in.

To facilitate the comparison, it can be helpful to convert all the measurements to the same denominator:

1 5/8in. = (8 + 5)/8 = 13/8in.
1 11/16in. = (16 + 11)/16 = 27/16in.
1 1 1 1 3/4in. = (4 x 16 + 3)/16 = 67/16in.
1 1/2in. = (2 + 1)/2 = 3/2in.

Now, we can determine which fraction is the largest.

Comparing the fractions, we can see that 67/16in. is the largest among them.

Therefore, the width of the widest stamp is 67/16in.
Answered by GPT 3.5
The circumference of a circle can be calculated using the formula C = πd, where C is the circumference and d is the diameter.

Given that the diameter is 4.5 miles, we can plug in this value into the formula:

C = 3.14 * 4.5
C = 14.13

Therefore, the circumference of the circle with a diameter of 4.5 miles is approximately 14.13 miles when rounded to the hundredths place.
Answered by GPT 3.5
To write 6,739 in expanded form, we break down the number by its place values.

6,739 can be written in expanded form as:

6,000 + 700 + 30 + 9

Therefore, the expanded form of 6,739 is:

6,000 + 700 + 30 + 9
Answered by GPT 3.5
To express 8/11 as a repeating decimal, we can perform long division:

0.7272...
_______________
11 | 8.0000
-7.92
_______
80
-77
_____
30
- 22
_____
80
- 77
_____
30
- 22
_____
80
- 77
_____
30
- 22
_____
80
- 77
_____
30
- 22
_____
...

We can see that the remainder repeats as 30-22-80-77. Therefore, the decimal representation of 8/11 is 0.7272... where the cycle of repeating digits is marked by a bar above the digits: 0.72̅.
Answered by GPT 3.5
The number 0.72 is already in simplest form because it does not contain any common factors that can be simplified.
Answered by GPT 3.5
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can let x = 0.727272... and write an equation to solve for x.

To do this, we can multiply both sides of the equation by 100 to get rid of the repeating decimal:

100x = 72.727272...

Next, we subtract x from both sides to eliminate the repeating decimal:

100x - x = 72.727272... - 0.727272...
99x = 72

Now, we can divide both sides by 99 to solve for x:

x = 72/99

To simplify the fraction 72/99, we can find the greatest common divisor (GCD) of the numerator and denominator, which is 9:

72/99 = (72/9) / (99/9) = 8/11

Therefore, 0.72¯¯¯¯¯ is equivalent to the rational number 8/11 in simplest form.
Answered by GPT 3.5
To find a rational number between 5 and 5 1/4, we can look for a fraction that falls between these two numbers on the number line.

5 can be written as 5/1, and 5 1/4 can be written as 21/4.

To find a rational number between these two fractions, we can think of a fraction with a numerator and denominator that lie between 5 and 21/4.

One possible rational number that fits this criteria is 19/4, which can be expressed as 4 3/4.

Therefore, the rational number 4 3/4 is between 5 and 5 1/4.
Answered by GPT 3.5
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we can take the average of these two numbers.

First, let's simplify the given numbers:

7–√ = 7 – √(7) = 7 – 2.645751 = 4.354249
8–√ = 8 – √(8) = 8 – 2.828427 = 5.171573

Now, we can find the average of these two numbers:

(4.354249 + 5.171573) / 2 = 9.525822 / 2 = 4.762911

To convert this decimal to a fraction with a denominator of 7, we multiply both the numerator and denominator of the fraction by 7:

4.762911 * 7 / 1 * 7 = 33.340377 / 7

Therefore, a rational number between 7–√ and 8–√, with a denominator of 7, is 33.340377 / 7 written as an improper fraction.
Answered by GPT 3.5
Apologies for the incorrect response. Let's try again.

To find a rational number with a denominator of 7 that falls between 7–√ and 8–√, we can look for a fraction that lies between these two radical expressions.

Let's simplify 7–√ and 8–√:

7–√ = 7 – √7
8–√ = 8 – √8 = 8 – 2√2

Now, let's find a rational number with a denominator of 7 that falls between these two expressions.

To do this, we can take the average of the two expressions:

(7 – √7 + 8 – 2√2) / 2 = (15 – √7 – 2√2) / 2

To simplify further, we can multiply the numerator and denominator by 2 to eliminate the fraction:

(2(15) – 2(√7) – 2(2√2)) / 2 = (30 – 2√7 – 4√2) / 2

Now, we can simplify the expression:

(30 / 2) – (2√7 / 2) – (4√2 / 2) = 15 – √7 – 2√2

Therefore, the rational number with a denominator of 7 that is between 7–√ and 8–√ is (15 – √7 – 2√2) / 7 written as an improper fraction.
Answered by GPT 3.5
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 fraction with a denominator of 1. For example, 3, 7, and -1 are all whole numbers, and they can be written as fractions: 3/1, 7/1, and -1/1.

On the other hand, not all rational numbers are whole numbers. Rational numbers include all integers, fractions, and terminating or repeating decimals. Examples of rational numbers that are not whole numbers include 1/2, -3/4, and 2.5.

In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. Whole numbers form a subsection within the larger category of rational numbers.