Which set of numbers is always rational?(1 point)

Question

Responses

decimals
decimals

positive numbers
positive numbers

integers
integers

negative numbers

Answer 1

The set of numbers that is always rational is integers.

Rational numbers are defined as numbers that can be expressed as the quotient or fraction of two integers (where the denominator is not zero). Since integers can be expressed as fractions (for example, 5 can be written as 5/1), they are always rational.

Decimals can be rational or irrational (for example, 0.5 is rational, but π is not). Positive and negative numbers can also include both rational and irrational numbers. Thus, the correct answer is integers.

Answer 2

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?

Answer 3

To find the height of an equilateral triangle with a side length of 8 meters using the given relationship between height and side length, we can use the formula:

\[ \text{Height} = \frac{\sqrt{3}}{2} \times \text{side length} \]

Substituting the given value for \(\sqrt{3}\) (which is approximately \(7/4\)) and the side length of 8 meters into the formula:

\[ \text{Height} = \frac{7/4}{2} \times 8 \]

Now, simplify \(\frac{7/4}{2}\):

\[ \frac{7/4}{2} = \frac{7}{4} \times \frac{1}{2} = \frac{7}{8} \]

Now, substitute this back into the height formula:

\[ \text{Height} = \frac{7}{8} \times 8 \]

Calculating this:

\[ \text{Height} = 7 \]

Thus, the approximate height of the equilateral triangle with a side length of 8 meters is 7 meters.

Answer 4

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?

Answer 5

To find the length of the diagonal of a square, we can use the formula:

\[ \text{Diagonal} = \sqrt{2} \times \text{side length} \]

Using the approximation \(1.4\) for \(\sqrt{2}\) and a side length of \(700\) feet, we can calculate the diagonal as follows:

\[ \text{Diagonal} = 1.4 \times 700 \]

Now, perform the multiplication:

\[ \text{Diagonal} = 1.4 \times 700 = 980 \]

Thus, the approximate length in feet from one corner to the opposite corner of the square park is 980 feet.

Answer 6

Using a number line, what whole number and rational number to the nearest half is 40−−√
between?(1 point)
40−−√
is between
and
.

Answer 7

To determine between which whole and rational numbers \( \sqrt{40} \) lies, we first need to calculate the approximate value of \( \sqrt{40} \).

The square root of 40 can be simplified as follows:

\[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10} \]

Now, to find \( \sqrt{40} \), we can approximate \( \sqrt{10} \). Since \( 3^2 = 9 \) and \( 4^2 = 16 \), we know \( \sqrt{10} \) is between 3 and 4.

A more precise approximation is \( \sqrt{10} \approx 3.162 \) (this can be found using a calculator or estimation) .

So:

\[ \sqrt{40} \approx 2 \times 3.162 \approx 6.324 \]

Now, we can identify the whole numbers and rational numbers to the nearest half that \( 6.324 \) is between:

• The whole numbers it is between are 6 and 7.
• The nearest half values it is between are 6.0 and 6.5.

Thus, \( \sqrt{40} \) is between 6 and 7 and also between 6.0 and 6.5.

Answer 8

Using rational approximations, what is an inequality statement comparing 81−−√
and 8–√8
? Enter <, >, or =.(1 point)
The inequality is: 81−−√

8–√8
.

Answer 9

To compare \( \sqrt{81} \) and \( 8 - \sqrt{8} \), we first calculate each expression:

1. Calculating \( \sqrt{81} \): \[ \sqrt{81} = 9 \]

2. Calculating \( 8 - \sqrt{8} \): We need to compute the approximate value of \( \sqrt{8} \). Since \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \), and using \( \sqrt{2} \approx 1.414 \): \[ \sqrt{8} \approx 2 \times 1.414 \approx 2.828 \] Now, substituting this back into \( 8 - \sqrt{8} \): \[ 8 - \sqrt{8} \approx 8 - 2.828 \approx 5.172 \]

Now we can compare the two values:

• \( \sqrt{81} = 9 \)
• \( 8 - \sqrt{8} \approx 5.172 \)

Since \( 9 > 5.172 \), we have:

\[ \sqrt{81} > 8 - \sqrt{8} \]

Thus, the inequality statement is:

\[ \sqrt{81} > 8 - \sqrt{8} \]

Answer 10

Convert 0.72 *repeating*
to a rational number in simplest form.

Answer 11

To convert \(0.72\) (with \(2\) repeating) to a rational number, we can set it equal to a variable and then manipulate the equation.

Let:

\[ x = 0.727272\ldots \]

First, we can multiply both sides of the equation by \(100\) to shift the decimal point two places to the right (since the repeating part has a length of 2):

\[ 100x = 72.727272\ldots \]

Now, we have two equations:

1. \( x = 0.727272\ldots \)
2. \( 100x = 72.727272\ldots \)

Next, we can subtract the first equation from the second:

\[ 100x - x = 72.727272\ldots - 0.727272\ldots \]

This simplifies to:

\[ 99x = 72 \]

Now, divide both sides by \(99\):

\[ x = \frac{72}{99} \]

To simplify this fraction, we can find the greatest common divisor (GCD) of \(72\) and \(99\).

The GCD of \(72\) and \(99\) is \(9\).

Now, divide both the numerator and the denominator by \(9\):

\[ \frac{72 \div 9}{99 \div 9} = \frac{8}{11} \]

Thus, the repeating decimal \(0.72\) (with \(2\) repeating) can be expressed as the rational number:

\[ \frac{8}{11} \]

Final Answer: \( \frac{8}{11} \)