A’Leila is building a pen for her pet donkey. The pen is a rectangle with one side

Question

measuring b yards and the adjacent side measuring a yards. A’Leila knows that
a=1/3b. Write two different expressions giving the perimeter of the donkey’s pen.

Answer 1

The perimeter \( P \) of a rectangle is given by the formula:

\[ P = 2(a + b) \]

Since we know that \( a = \frac{1}{3}b \), we can substitute \( a \) into the perimeter formula.

Expression 1: Using the relationship \( a = \frac{1}{3}b \):

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

Expression 2: Alternatively, we can express the perimeter directly in terms of \( a \). Since \( a = \frac{1}{3}b \), we can express \( b \) in terms of \( a \):

\[ b = 3a \]

Now substitute \( b = 3a \) into the perimeter formula:

\[ P = 2(a + b) \] \[ P = 2\left(a + 3a\right) \] \[ P = 2(4a) \] \[ P = 8a \]

So, the two different expressions for the perimeter of the donkey’s pen are:

1. \( P = \frac{8}{3}b \)
2. \( P = 8a \)