A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side measuring b yards and the adjacent side measuring a yards. A’Leila knows that a = 1/3b.

a. To find the perimeter \( P \) of a rectangle, the formula is given by:

\[ P = 2a + 2b \]

Since A’Leila knows that \( a = \frac{1}{3}b \), we can express \( b \) in terms of \( a \):

\[ b = 3a \]

Now we can write a second expression for the perimeter using this substitution:

1. Using the original dimensions: \[ P = 2a + 2b \]

2. Using the relationship between \( a \) and \( b \): Substituting \( b = 3a \) into the perimeter formula: \[ P = 2a + 2(3a) = 2a + 6a = 8a \]

So, the two expressions for the perimeter are:

1. \( P = 2a + 2b \)
2. \( P = 8a \)

b. Let's choose a value for \( a \): Let \( a = 6 \) (which is greater than 5 and less than 15).

Now, we can evaluate the perimeter using the second expression \( P = 8a \):

\[ P = 8(6) = 48 \text{ yards} \]

c. I chose the expression \( P = 8a \) to evaluate the perimeter of the donkey’s pen because it provides a direct relationship between the perimeter and the measurement of one side of the rectangle, \( a \). Since the problem specifies that \( a \) can be conveniently chosen, this expression simplifies the calculation and eliminates the need to calculate \( b \) before finding the perimeter. This makes it easier to quickly determine the perimeter based on the chosen value of \( a \).