7. A square patio has an area of 200 square feet. How long is each side of the patio to the nearest tenth? 1.2 Sets of Real Numbers Write all names that apply to each number. 8. $$ \frac{121}{\sqrt{121}} $$ 9. $$ \frac{\pi}{2} $$ 10. Tell whether the statement "All integers are rational numbers" is true or false. Explain your choice.

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Sure, let's tackle each of these problems one by one.

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**7. A square patio has an area of 200 square feet. How long is each side of the patio to the nearest tenth?**

To find the length of each side of a square when given the area:

1. **Area formula for a square:** 
   $$
   \text{Area} = (\text{side length})^2
   $$
   
2. **Given:** 
   $$
   (\text{side length})^2 = 200
   $$
   
3. **Solve for the side length:**
   $$
   \text{side length} = \sqrt{200}
   $$
   
4. **Calculate the square root:**
   $$
   \sqrt{200} \approx 14.1421 \text{ feet}
   $$
   
5. **Round to the nearest tenth:**
   $$
   14.1421 \approx 14.1 \text{ feet}
   $$
   
**Answer:** Each side of the patio is approximately **14.1 feet** long.

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**1.2 Sets of Real Numbers**

*Write all names that apply to each number.*

**8. $$ \frac{121}{\sqrt{121}} $$**

First, simplify the expression:

$$
\frac{121}{\sqrt{121}} = \frac{121}{11} = 11
$$

Now, determine the types of numbers 11 belongs to:

- **Natural Numbers (N):** Yes, 11 is a natural number (counting number).
- **Whole Numbers (W):** Yes, it is a whole number (natural numbers including zero).
- **Integers (Z):** Yes, it is an integer (whole numbers and their negatives).
- **Rational Numbers (Q):** Yes, since it can be expressed as $$ \frac{11}{1} $$.
- **Irrational Numbers:** No, it is not irrational.
- **Real Numbers (R):** Yes, as all rational numbers are real numbers.
- **Other Classifications:**
  - It is a **positive** number.
  - It is a **whole** positive integer.
  
**Answer:** The number 11 is a **natural number, whole number, integer, rational number, and real number**.

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**9. $$ \frac{\pi}{2} $$**

Determine the types of numbers $$ \frac{\pi}{2} $$ belongs to:

- **Natural Numbers (N):** No, it's not a counting number.
- **Whole Numbers (W):** No, it's not a whole number.
- **Integers (Z):** No, it is not an integer.
- **Rational Numbers (Q):** No, because $$ \pi $$ is irrational, and dividing by 2 doesn't make it rational.
- **Irrational Numbers:** Yes, $$ \pi $$ is irrational, and dividing by 2 maintains its irrationality.
- **Real Numbers (R):** Yes, all irrational numbers are real numbers.
  
**Answer:** The number $$ \frac{\pi}{2} $$ is an **irrational number and a real number**.

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**10. Tell whether the statement "All integers are rational numbers" is true or false. Explain your choice.**

**Statement:** *All integers are rational numbers.*

**Answer:** **True.**

**Explanation:** 
A rational number is defined as any number that can be expressed as the quotient or fraction $$ \frac{p}{q} $$, where $$ p $$ and $$ q $$ are integers and $$ q \neq 0 $$.

Since any integer $$ z $$ can be written as $$ \frac{z}{1} $$, it fits the definition of a rational number. For example, the integer 5 can be expressed as $$ \frac{5}{1} $$, and -3 as $$ \frac{-3}{1} $$.

Therefore, **all integers are indeed rational numbers**.

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**7. Each side of the patio is approximately 14.1 feet long.** **1.2 Sets of Real Numbers** **8. $$ \frac{121}{\sqrt{121}} $$ is a natural number, whole number, integer, rational number, and real number.** **9. $$ \frac{\pi}{2} $$ is an irrational number and a real number.** **10. The statement "All integers are rational numbers" is **true** because every integer can be expressed as a fraction with a denominator of 1, satisfying the definition of a rational number.

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