To solve this question, we need to understand the expressions for width and length and determine both the mathematical and reasonable domains for the variable [tex]\( x \)[/tex].
Given the expressions for width and length:
- Width: [tex]\( 0.5x - 2 \)[/tex]
- Length: [tex]\( 2x + 2 \)[/tex]
### Step 1: Identify the mathematical domain
The mathematical domain is the set of all possible values of [tex]\( x \)[/tex] for which the expressions are defined. The expressions [tex]\( 0.5x - 2 \)[/tex] and [tex]\( 2x + 2 \)[/tex] are both linear functions, which means they are defined for all real numbers. Hence, the mathematical domain is:
[tex]\[ \{x \in \mathbb{R}\} \][/tex]
### Step 2: Define the constraints for the reasonable domain
Next, we need to find the range of [tex]\( x \)[/tex] where the dimensions make sense. Specifically, the width and the length must be positive, and the length should not exceed 52 cm.
#### Constraint 1: Width is positive
[tex]\[ 0.5x - 2 > 0 \][/tex]
[tex]\[ 0.5x > 2 \][/tex]
[tex]\[ x > 4 \][/tex]
#### Constraint 2: Length does not exceed 52 cm
[tex]\[ 2x + 2 \leq 52 \][/tex]
[tex]\[ 2x \leq 50 \][/tex]
[tex]\[ x \leq 25 \][/tex]
### Step 3: Combine the constraints for the reasonable domain
We combine these two constraints:
[tex]\[ 4 < x \leq 25 \][/tex]
Thus, the reasonable domain is:
[tex]\[ \{4 < x \leq 25\} \][/tex]
### Summary
- The mathematical domain is:
[tex]\[ \{x \in \mathbb{R}\} \][/tex]
- The reasonable domain is:
[tex]\[ \{4 < x \leq 25\} \][/tex]
Therefore, the option that matches these findings is:
[tex]\[ \text{mathematical: } \{x \in \mathbb{R}\} \quad \text{reasonable: } \{4 < x \leq 25\} \][/tex]
