Question 4:

To find the perimeter of A’Leila's rectangular pen, we can use the formula for the perimeter of a rectangle:

\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]

In this case, the sides of the rectangle are "a" (width) and "b" (length).

Part a:

We know from the problem statement that \( a = \frac{1}{3}b \).

1. First Expression: Using the formula for perimeter in terms of \( b \):

   \[ \text{Perimeter} = 2 \times (a + b) \] Substituting \( a = \frac{1}{3}b \):

   \[ \text{Perimeter} = 2 \times \left(\frac{1}{3}b + b\right) \] This simplifies to:

   \[ \text{Perimeter} = 2 \times \left(\frac{1}{3}b + \frac{3}{3}b\right) = 2 \times \left(\frac{4}{3}b\right) = \frac{8}{3}b \]

2. Second Expression: Alternatively, we can express the perimeter by substituting directly into the perimeter formula:

   We can rewrite it fully in terms of \( a \):

   \[ \text{Perimeter} = 2 \times (a + b) = 2a + 2b \]

   Since \( a = \frac{1}{3}b \):

   \[ \text{Perimeter} = 2\left(\frac{1}{3}b\right) + 2b \] This simplifies to:

   \[ \text{Perimeter} = \frac{2}{3}b + 2b = \frac{2}{3}b + \frac{6}{3}b = \frac{8}{3}b \]

Both expressions are equivalent, so our two expressions for the perimeter in terms of \( b \) are:

1. \( \frac{8}{3}b \)
2. \( 2a + 2b \) (which simplifies to \( \frac{8}{3}b \) when \( a \) is substituted)

Part b:

Now, using \( a = 12 \) to evaluate the perimeter using one of our expressions:

From \( a = \frac{1}{3}b \), we can find \( b \).

Since \( a = 12 \):

\[ 12 = \frac{1}{3}b \] Multiplying both sides by 3:

\[ b = 36 \]

Now, we can substitute \( b \) into the perimeter expression:

Using \( \text{Perimeter} = \frac{8}{3}b \):

\[ \text{Perimeter} = \frac{8}{3} \times 36 \]

Calculating that:

\[ \text{Perimeter} = \frac{8 \times 36}{3} = \frac{288}{3} = 96 \]

Therefore, the perimeter of the pen when \( a = 12 \) is:

96 yards.