To find the perimeter of A’Leila's donkey pen, which is a rectangle with sides \( a \) and \( b \), we can use the geometric formula for the perimeter of a rectangle given by:
\[ P = 2(a + b) \]
Given that \( a = \frac{1}{3}b \), we can express the perimeter in terms of \( b \).
a. Expressions for Perimeter using Variable “b” ONLY
- Substitute \( a \) with \( \frac{1}{3}b \) in the perimeter formula:
\[ P = 2\left( \frac{1}{3}b + b \right) \]
Now, combine the terms inside the parentheses:
\[ P = 2\left( \frac{1}{3}b + \frac{3}{3}b \right) = 2\left( \frac{4}{3}b \right) = \frac{8}{3}b \]
- Another way to express the perimeter is to write it as a function of \( b \) later. Since we have \( a = \frac{1}{3}b \), we can also represent it as follows:
\[ P = 2a + 2b \]
Now, substituting \( a \):
\[ P = 2\left(\frac{1}{3}b\right) + 2b = \frac{2}{3}b + 2b = \frac{2}{3}b + \frac{6}{3}b = \frac{8}{3}b \]
Both expressions give the same perimeter:
Expressions:
- \( P = \frac{8}{3}b \)
- \( P = 2\left(\frac{1}{3}b + b\right) = \frac{8}{3}b \)
b. Evaluating the Perimeter using \( a = 12 \)
Given \( a = 12 \), we can first determine \( b \):
Since \( a = \frac{1}{3}b \):
\[ 12 = \frac{1}{3}b \]
Multiplying both sides by 3:
\[ b = 36 \]
Now we can use one of the perimeter expressions to find \( P \):
Using the expression \( P = \frac{8}{3}b \):
\[ P = \frac{8}{3}(36) \]
Calculating \( P \):
\[ P = \frac{8 \times 36}{3} = \frac{288}{3} = 96 \]
Therefore, the perimeter of the pen when \( a = 12 \) is:
Perimeter: \( P = 96 \) yards.