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

Question

Responses

positive numbers
positive numbers

negative numbers
negative numbers

integers
integers

decimals

Answer 1

The set of numbers that is always rational is integers.

Integers are whole numbers that can be positive, negative, or zero, and they can be expressed as fractions with a denominator of 1 (e.g., 3 can be expressed as 3/1).

Positive numbers, negative numbers, and decimals can include rational and irrational numbers, whereas integers are always rational.

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, we can use the formula:

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

Given that the side length is 8 meters and using \( \sqrt{3} \approx \frac{7}{4} \), we can substitute these values into the formula:

1. First, substitute the value for the square root of 3:

\[ \text{Height} \approx \frac{7}{4} \times \frac{1}{2} \times 8 \]

2. Simplifying this step by step:

\[ \text{Height} \approx \frac{7}{4} \times 4 \]

3. The 4's cancel out:

\[ \text{Height} \approx 7 \text{ meters} \]

So, 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 diagonal of a square, we can use the formula:

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

Given that the side length of the square park is 700 feet and using \( \sqrt{2} \approx 1.4 \), we can substitute these values into the formula:

1. Substitute the values:

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

2. Calculate the diagonal:

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

Therefore, the approximate length 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

Answer 7

To find between which whole number and rational number (to the nearest half) the value of \( \sqrt{40} \) lies, we first need to compute \( \sqrt{40} \).

1. Simplifying \( \sqrt{40} \):

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

Next, we can approximate \( \sqrt{10} \). The value of \( \sqrt{10} \) is approximately 3.162. Thus,

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

2. Now, we will determine the whole number and the nearest half. The whole number parts surrounding 6.324 are:

• The largest whole number less than 6.324 is 6.
• The smallest whole number greater than 6.324 is 7.

3. To find the nearest half, we compare 6.324 to .0 and .5:

• It is closer to 6.5 than to 6.0.

Conclusion:

• The whole number is 6.
• The rational number to the nearest half is 6.5.

Therefore, \( \sqrt{40} \) is between 6 and 6.5.

Answer 8

Using a number line, what whole number and rational number to the nearest half is √58
between

Answer 9

To find between which whole number and rational number (to the nearest half) \( \sqrt{58} \) lies, we can start by calculating \( \sqrt{58} \).

1. Estimate \( \sqrt{58} \):

   We know that:

   • \( \sqrt{49} = 7 \)
   • \( \sqrt{64} = 8 \)

   Since \( 58 \) is between \( 49 \) and \( 64 \), we can conclude that:

   \[ 7 < \sqrt{58} < 8 \]

2. Calculate \( \sqrt{58} \) more accurately:

   Using a calculator, we find:

   \[ \sqrt{58} \approx 7.6158 \]

3. Determine the whole number and rational number to the nearest half:

   • The largest whole number less than \( 7.6158 \) is 7.
   • The smallest whole number greater than \( 7.6158 \) is 8.

To find the nearest half:

• \( 7.6158 \) is closer to \( 7.5 \) than to \( 8.0 \).

Conclusion:

• The whole number is 7.
• The rational number to the nearest half is 7.5.

Thus, \( \sqrt{58} \) is between 7 and 7.5.