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

Question

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

positive numbers
positive numbers

negative numbers
negative numbers

decimals
decimals

integers

GPT-4o mini · Answer

The set of numbers that is always rational is integers.

Rational numbers are defined as numbers that can be expressed as a fraction or ratio of two integers (where the denominator is not zero). Integers (which include positive numbers, negative numbers, and zero) fit this definition because any integer can be expressed as a fraction with a denominator of 1 (e.g., 5 can be written as 5/1). Thus, all integers are rational numbers.

The other options (positive numbers, negative numbers, and decimals) can include irrational numbers (for example, √2 is a positive irrational number), so they are not guaranteed to be rational.

Answer

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)
$$

GPT-4o mini · Answer

To calculate the height of an equilateral triangle given that the height $ h $ equals $ 3\sqrt{2} $ times the side length $ s $, we can follow these steps:

Given:
- $ s = 8 $ meters
- $ h = 3\sqrt{2} \cdot s $
Substituting the value of $ s $: \[ h = 3\sqrt{2} \cdot 8 \]
Calculate $ 3 \cdot 8 $: \[ 3 \cdot 8 = 24 \] Thus, \[ h = 24\sqrt{2} \]
Now substituting $ \sqrt{2} $: Since the problem asks for the approximation and does not provide a numerical value for $ \sqrt{2} $, we will use the approximation $ \sqrt{2} \approx 1.414 $ for calculation. Thus, \[ h \approx 24 \cdot 1.414 \]
Calculating $ 24 \cdot 1.414 $: \[ h \approx 24 \cdot 1.414 \approx 33.936 \]
Final height, rounded to two decimal places: \[ h \approx 33.94 \text{ meters} \]

Therefore, the approximate height of the equilateral triangle with a side length of 8 meters is about 33.94 meters.

Answer

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?(1 point)
$$

GPT-4o mini · Answer

To find the length of the diagonal $ d $ of a square given that the diagonal equals $ \sqrt{2} $ times the side length $ s $, we can follow these steps: 1. **Given:** - The side length $ s = 700 $ feet - The relationship between the diagonal and the side length is $ d = \sqrt{2} \cdot s $ 2. **Substituting the value of $ s $:** $$ d = \sqrt{2} \cdot 700 $$ 3. **Using the provided approximation for $ \sqrt{2} $:** - We will use $ \sqrt{2} \approx 1.4 $. 4. **Calculating the diagonal:** $$ d \approx 1.4 \cdot 700 $$ 5. **Calculating $ 1.4 \cdot 700 $:** $$ d \approx 1.4 \cdot 700 = 980 $$ Therefore, the approximate length from one corner to the opposite corner of the square park that is 700 feet on a side is **980 feet**.

Answer

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

GPT-4o mini · Answer

To determine the whole number and the rational number to the nearest half that $ \sqrt{40} $ is between, we first need to calculate $ \sqrt{40} $.

Calculate $ \sqrt{40} $: \[ \sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10} \]
Estimate $ \sqrt{10} $: We know that:
- $ \sqrt{9} = 3 $ (since $ 3^2 = 9 $)
- $ \sqrt{10} $ is slightly more than 3.
A more precise estimate for $ \sqrt{10} $ using a calculator gives approximately $ 3.162 $.
Calculate $ \sqrt{40} $: \[ \sqrt{40} \approx 2 \cdot 3.162 \approx 6.324 \]
Determine the whole number: The whole number just below $ 6.324 $ is $ 6 $, and the whole number just above it is $ 7 $.
Determine the rational number to the nearest half: The nearest half to $ 6.324 $ is $ 6.5 $.

Now, we can conclude that:

\[ \sqrt{40} \text{ is between } 6 \text{ and } 6.5. \]

So the final answer is:

$ \sqrt{40} $ is between 6 and 6.5.

Answer

Using a number line, what whole number and rational number to the nearest half is 58−−√
between?(1 point)
Responses

between 8 and 8.5
between 8 and 8.5

between 7.5 and 8
between 7.5 and 8

between 8.5 and 9
between 8.5 and 9

between 7 and 7.5

GPT-4o mini · Answer

To determine where $ \sqrt{58} $ falls on the number line and which whole number and rational number (to the nearest half) it is between, we first need to approximate $ \sqrt{58} $. 1. **Estimate $ \sqrt{58} $:** - We know that: - $ \sqrt{49} = 7 $ (since $ 7^2 = 49 $) - $ \sqrt{64} = 8 $ (since $ 8^2 = 64 $) Since $ 58 $ is between $ 49 $ and $ 64 $, we can deduce that: $$ 7 < \sqrt{58} < 8 $$ 2. **Calculate $ \sqrt{58} $ more precisely:** A more precise estimate can be found using a calculator: $$ \sqrt{58} \approx 7.615 $$ 3. **Determine the whole number:** From our estimation, the whole number just below $ 7.615 $ is $ 7 $, and the whole number above it is $ 8 $. 4. **Determine the rational number to the nearest half:** The number $ 7.615 $ rounds to the nearest half as $ 7.5 $. Putting this all together, we find that: $$ \sqrt{58} \text{ is between } 7.5 \text{ and } 8. $$ Thus, the correct response is: **between 7.5 and 8.**

Answer

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

GPT-4o mini · Answer

To compare $ \sqrt{81} $ and $ 8 - \sqrt{8} $, let's calculate each expression step by step. 1. **Calculate $ \sqrt{81} $:** $$ \sqrt{81} = 9 $$ 2. **Estimate $ \sqrt{8} $:** We know that: - $ \sqrt{4} = 2 $ - $ \sqrt{9} = 3 $ Since $ 8 $ is between $ 4 $ and $ 9 $, we can estimate that $ \sqrt{8} $ is between $ 2 $ and $ 3 $. A more precise value for $ \sqrt{8} $ is approximately $ 2.828 $ (as $ \sqrt{8} = 2\sqrt{2} $). 3. **Calculate $ 8 - \sqrt{8} $:** $$ 8 - \sqrt{8} \approx 8 - 2.828 \approx 5.172 $$ 4. **Compare $ \sqrt{81} $ and $ 8 - \sqrt{8} $:** Now we compare: - $ \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: $$ \sqrt{81} > 8 - \sqrt{8} $$ So the answer is: **>**