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The expression represents the perimeter of the rectangular pen if [tex]\(a\)[/tex] is the width of the pen.

Given:
- Width [tex]\(= a\)[/tex]
- Length [tex]\(= 2a\)[/tex]

Perimeter [tex]\(= 2 \times (\text{length} + \text{width}) = 2 \times (2a + a) = 6a\)[/tex]

If the variable [tex]\(a\)[/tex] represents the length of the rectangular pen in the same example:

- Width [tex]\(= \frac{a}{2}\)[/tex]

Using the formula for perimeter:
[tex]\[
\begin{array}{l}
\text{Perimeter of rectangle} = 2 \times (\text{length} + \text{width}) \\
= 2 \left(a + \frac{a}{2}\right) \\
= 2 \left(\frac{3a}{2}\right) \\
= 3a
\end{array}
\][/tex]

Note: There seems to be a mismatch in the provided information; if you intended to use [tex]\(\epsilon\)[/tex] or another variable, please ensure it's correctly defined in the context.



Answer :

GinnyAnswer
Let's analyze the problem step by step and simplify the expressions:

Given:

1. The length of the rectangle is [tex]\( a \)[/tex].
2. The width of the rectangle is [tex]\( \frac{a}{2} \)[/tex].

To find the perimeter of the rectangle, we use the formula for the perimeter of a rectangle:

[tex]\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \][/tex]

Substitute the given length and width into the formula:

[tex]\[ \text{Perimeter} = 2 \times \left(a + \frac{a}{2}\right) \][/tex]

Simplify inside the parentheses:

[tex]\[ a + \frac{a}{2} = \frac{2a}{2} + \frac{a}{2} = \frac{3a}{2} \][/tex]

So, the expression becomes:

[tex]\[ \text{Perimeter} = 2 \times \frac{3a}{2} \][/tex]

The 2's cancel each other out:

[tex]\[ \text{Perimeter} = 3a \][/tex]

Thus, the perimeter of the rectangle in simplified form is [tex]\( 3a \)[/tex].

Using the given values, let's validate the calculations step by step:

- The width is [tex]\( \frac{a}{2} = 1.0 \)[/tex].
- The original formula we found the perimeter using is [tex]\( 2 \times \left(a + \frac{a}{2}\right) = 2 \times 1.5 \times a = 3 \times a \)[/tex].
- The simplified expression for the perimeter is [tex]\( 3a \)[/tex], which, if [tex]\( a = 2 \)[/tex], gives [tex]\( 3 \times 2 = 6 \)[/tex].

Thus, the width is [tex]\( 1 \)[/tex], and the perimeter of the rectangular pen with [tex]\( a = 2 \)[/tex] is [tex]\( 6 \)[/tex].

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