Answer :
Let's determine the length and width of the rectangular garden whose area is represented by the expression \( r^2 + 11r + 18 \).
### Step 1: Understand the problem
The area of a rectangle is given by the product of its length and width. Thus, we are looking to express \( r^2 + 11r + 18 \) in the form of \( (r + a)(r + b) \), where \( a \) and \( b \) are constants.
### Step 2: Factor the quadratic expression
To factorize \( r^2 + 11r + 18 \), we need to find two numbers that:
1. Multiply to give 18 (the constant term), and
2. Add to give 11 (the coefficient of the linear term, \( r \)).
### Step 3: Find the factors
We look for pairs of numbers that multiply to 18:
- \( 1 \times 18 = 18 \)
- \( 2 \times 9 = 18 \)
- \( 3 \times 6 = 18 \)
Next, we check which of these pairs add to 11:
- \( 1 + 18 = 19 \)
- \( 2 + 9 = 11 \)
- \( 3 + 6 = 9 \)
The pair \( 2 \) and \( 9 \) add up to 11. Therefore, the correct factorization of the quadratic expression is:
[tex]\[ r^2 + 11r + 18 = (r + 2)(r + 9) \][/tex]
### Step 4: Verify the factors
Let's quickly verify the factorization by expanding \( (r + 2)(r + 9) \):
[tex]\[ (r + 2)(r + 9) = r^2 + 9r + 2r + 18 = r^2 + 11r + 18 \][/tex]
This confirms that the factorization is correct.
### Step 5: Identify the length and width
From the factorized form \( (r + 2)(r + 9) \), we can see:
- One dimension is \( r + 2 \)
- The other dimension is \( r + 9 \)
### Step 6: Check the options
Given the options:
A. \( (r + 18), (r + 1) \)
B. \( (r + 11), (r + 18) \)
C. \( (r + 6), (r + 3) \)
D. none of these
None of these options correctly represent the factorized form \( (r + 2) \) and \( (r + 9) \).
### Conclusion
The correct answer is:
D. none of these
### Step 1: Understand the problem
The area of a rectangle is given by the product of its length and width. Thus, we are looking to express \( r^2 + 11r + 18 \) in the form of \( (r + a)(r + b) \), where \( a \) and \( b \) are constants.
### Step 2: Factor the quadratic expression
To factorize \( r^2 + 11r + 18 \), we need to find two numbers that:
1. Multiply to give 18 (the constant term), and
2. Add to give 11 (the coefficient of the linear term, \( r \)).
### Step 3: Find the factors
We look for pairs of numbers that multiply to 18:
- \( 1 \times 18 = 18 \)
- \( 2 \times 9 = 18 \)
- \( 3 \times 6 = 18 \)
Next, we check which of these pairs add to 11:
- \( 1 + 18 = 19 \)
- \( 2 + 9 = 11 \)
- \( 3 + 6 = 9 \)
The pair \( 2 \) and \( 9 \) add up to 11. Therefore, the correct factorization of the quadratic expression is:
[tex]\[ r^2 + 11r + 18 = (r + 2)(r + 9) \][/tex]
### Step 4: Verify the factors
Let's quickly verify the factorization by expanding \( (r + 2)(r + 9) \):
[tex]\[ (r + 2)(r + 9) = r^2 + 9r + 2r + 18 = r^2 + 11r + 18 \][/tex]
This confirms that the factorization is correct.
### Step 5: Identify the length and width
From the factorized form \( (r + 2)(r + 9) \), we can see:
- One dimension is \( r + 2 \)
- The other dimension is \( r + 9 \)
### Step 6: Check the options
Given the options:
A. \( (r + 18), (r + 1) \)
B. \( (r + 11), (r + 18) \)
C. \( (r + 6), (r + 3) \)
D. none of these
None of these options correctly represent the factorized form \( (r + 2) \) and \( (r + 9) \).
### Conclusion
The correct answer is:
D. none of these
