Which of the following is true about −9?(1 point)
Responses
It is both an integer and a whole number.
It is both an integer and a whole number.
It is an integer but not a whole number.
It is an integer but not a whole number.
It is an integer but not a rational number.
It is an integer but not a rational number.
It is a whole number but not an integer.
It is an integer but not a whole number.
A sign in a gas station advertises gas at the rate of $3.39910 per gallon. How many dollars does a gallon cost, writing the price strictly as a decimal?
A gallon costs $3.39910.
Which set of numbers is always rational?(1 point)
Responses
positive numbers
positive numbers
decimals
decimals
fractions
fractions
integers
integers
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?
The height of an equilateral triangle equals 3√2 times the side length. Using 74 for the square root of 3, we can calculate the approximate height.
Height = 3√2 * side length
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.
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?
The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, we can calculate the approximate length from one corner to an opposite corner of a square park that is 700 feet on a side.
Diagonal = 2–√ * side length
Diagonal = 2(1.4) * 700
Diagonal = 2.8 * 700
Diagonal = 1960
Therefore, the approximate length from one corner to an opposite corner of the square park is 1960 feet.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
To determine the whole number and rational number to the nearest half that 40−−√ is between, we can use a number line.
The square root of 40 is approximately 6.32. To find the whole number and rational number that is closest, we can locate 6.32 on the number line and identify the nearest half.
On the number line, 6.32 will fall between the whole numbers 6 and 7.
To find the rational number closest to 6.32, we look at the midpoint between 6 and 7, which is 6.5.
Therefore, 40−−√ is between the whole number 6 and the rational number 6.5 on the number line.
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar
To express 811 as a repeating decimal, we need to perform long division.
The long division of 811 by 9 is as follows:
______
9 | 811.000
- 729
______
81
- 81
______
0
Since there is no remainder, the division terminates, and 811 can be expressed as a terminating decimal.
So, 811 as a decimal is 811.000.
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'll use the property of repeating decimals.
Let's represent 0.72¯¯¯¯¯ as x:
x = 0.727272...
Now, we want to eliminate the repeating part by shifting the decimal point. We can achieve this by multiplying x by a power of 10 that is equal to the number of repeating digits.
Since there are 2 repeating digits (72), we'll multiply x by 100:
100x = 72.727272...
Next, we subtract the original equation from the equation multiplied by 100:
100x - x = 72.727272... - 0.727272...
99x = 72
To isolate x, we divide both sides of the equation by 99:
x = 72/99
We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 9:
x = (8 * 9) / (11 * 9)
x = 8/11
Therefore, the rational form of the repeating decimal 0.72¯¯¯¯¯ is 8/11.
Which of the following is a rational number between 5 and 514
To determine a rational number between 5 and 514, we can choose any fraction with a numerator and denominator that yield a value between 5 and 514.
For example, we can choose the fraction 1/100, which is equivalent to 0.01. This fraction lies between 5 and 514.
Therefore, 1/100 is a rational number between 5 and 514.
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 lies between 7–√ and 8–√, we can find their average.
The average of the two numbers can be calculated as:
(7–√ + 8–√) / 2
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator to eliminate the square root in the numerator:
((7–√) * (7+√) + (8–√) * (7+√)) / (2 * (7+√))
Simplifying this expression further, we get:
(49 - (√ * √) + 7√ + 7√ + 56 - (√ * √) + 8√ - √) / (14 + 2√)
Combining like terms, we have:
(49 + 56 + 14 - 2√) / (14 + 2√)
Simplifying this expression gives:
(119 - 2√) / (14 + 2√)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator, which is (14 - 2√):
((119 - 2√) * (14 - 2√)) / ((14 + 2√) * (14 - 2√))
Expanding this expression, we get:
(1666 - 238√ - 28√ + 4√^2) / (196 - 4√^2)
Simplifying further, we have:
(1666 - 266√ + 4 * 2) / (196 - 4 * 2)
Now, combining like terms, we get:
(1674 - 266√) / 188
Therefore, a rational number with a denominator of 7 that lies between 7–√ and 8–√ is:
(1674 - 266√) / 188 ≈ 8.914893617
As an improper fraction, it can be expressed as:
(1674 - 266√) / 188 = (837 - 133√) / 94