Let's break this down step-by-step to explain what happens if the variable [tex]\( a \)[/tex] represents the length of a rectangular pen.
1. Understanding the Relationships:
- Suppose [tex]\( a \)[/tex] represents the length of the rectangular pen.
- The width of the pen is given as [tex]\( \frac{a}{2} \)[/tex].
2. Formulating the Perimeter:
- The formula for the perimeter of a rectangle is [tex]\( 2(\text{length} + \text{width}) \)[/tex].
- With [tex]\( a \)[/tex] as the length and [tex]\( \frac{a}{2} \)[/tex] as the width, substitute these values into the formula:
[tex]\[
\text{Perimeter} = 2(a + \frac{a}{2})
\][/tex]
3. Simplifying the Expression:
- Inside the parentheses, add the length and width:
[tex]\[
a + \frac{a}{2} = \frac{2a}{2} + \frac{a}{2} = \frac{2a + a}{2} = \frac{3a}{2}
\][/tex]
- Now, multiply by 2 to find the perimeter:
[tex]\[
2 \left(\frac{3a}{2}\right) = 3a
\][/tex]
4. Conclusion:
- The width of the pen when the length is [tex]\( a \)[/tex] is [tex]\( \frac{a}{2} \)[/tex].
- The perimeter of the pen is [tex]\( 3a \)[/tex].
Given these steps, the key results are:
- The width is half of the length, or [tex]\( \frac{a}{2} \)[/tex].
- The expression representing the perimeter of the rectangular pen if [tex]\( a \)[/tex] is the length is [tex]\( 3a \)[/tex].
