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

Given:
- Length [tex]\(= 2a\)[/tex]
- Perimeter [tex]\(= 68\)[/tex]
- Width [tex]\(= a\)[/tex]

What happens if the variable [tex]\(a\)[/tex] represents the length of the rectangular pen in the same example?

- The width is [tex]\(\frac{a}{2}\)[/tex].

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

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

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



Answer :

GinnyAnswer
To solve the question regarding the perimeter of a rectangular pen with the given conditions, let's proceed step-by-step.

### Given:
1. Length = [tex]\( 2a \)[/tex] (twice the width)
2. Perimeter = 68
3. Width = [tex]\( a \)[/tex]

### According to the formula for the perimeter of a rectangle:
[tex]\[ P = 2(\text{Length} + \text{Width}) \][/tex]

### Substitute the known values:
[tex]\[ 68 = 2(2a + a) \][/tex]

### Simplify the equation inside the parentheses:
[tex]\[ 68 = 2 \cdot 3a \][/tex]
[tex]\[ 68 = 6a \][/tex]

### Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{68}{6} \][/tex]
[tex]\[ a = \frac{34}{3} \][/tex]
[tex]\[ a \approx 11.\overline{3} \][/tex]

### Given:
- Width = [tex]\( a \approx 11.\overline{3} \)[/tex]
- Length = [tex]\( 2a = 2 \times 11.\overline{3} = 22.\overline{6} \)[/tex]

### Verification of the Perimeter:
Substitute the length and width back into the perimeter formula to confirm:
[tex]\[ \text{Perimeter} = 2(\text{Length} + \text{Width}) \][/tex]
[tex]\[ \text{Perimeter} = 2(22.\overline{6} + 11.\overline{3}) \][/tex]
[tex]\[ \text{Perimeter} = 2 \times 34 = 68 \][/tex]
This is correct.

### Now, if [tex]\( a \)[/tex] represents the length of the rectangular pen:
Given:
1. Width = [tex]\( \frac{a}{2} \)[/tex]
2. Length = [tex]\( a \)[/tex]
3. Perimeter formula:

[tex]\[ P = 2(\text{Length} + \text{Width}) \][/tex]
[tex]\[ P = 2\left(a + \frac{a}{2}\right) \][/tex]
[tex]\[ P = 2\left(\frac{3a}{2}\right) \][/tex]
[tex]\[ P = 3a \][/tex]

Given that the perimeter is the same:
[tex]\[ 3a = 68 \][/tex]

Solve for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{68}{3} \][/tex]
[tex]\[ a = 22.\overline{6} \][/tex]

So, when [tex]\( a \)[/tex] represents the length:
- Length = [tex]\( a \approx 22.\overline{6} \)[/tex]
- Width = [tex]\( \frac{a}{2} \approx \frac{22.\overline{6}}{2} = 11.\overline{3} \)[/tex]

### Conclusion:
When [tex]\( a \)[/tex] represents the length of the pen:
- The width is [tex]\( \frac{a}{2} \)[/tex].
- Using the formula for the perimeter, the relation [tex]\( P = 3a \)[/tex] explains that if the perimeter is 68, then the length [tex]\( a \)[/tex] would be [tex]\( 22.\overline{6} \)[/tex].

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