### 1. Relationship Between Sides

Question

You have:
- a = 1/3 b

### 2. Perimeter Formula
The formula for the perimeter P of a rectangle is:
P = a + b + a + b
This simplifies to:
P = 2a + 2b

### 3. Substitute a into the Perimeter Formula
Now we will substitute a with (1/3)b:
P = 2(1/3 b) + 2b

### 4. Simplify the Perimeter Formula
- First, calculate 2(1/3 b):
P = (2/3)b + 2b

- Now convert 2b to a fraction:
2b = (6/3)b

- Now combine the two terms:
P = (2/3)b + (6/3)b
P = (8/3)b

### 5. Finding b Given a = 10
Now, substitute a = 10 into the relationship:
10 = (1/3)b

- Multiply both sides by 3:
b = 10 * 3
b = 30

### 6. Calculate the Perimeter Using b = 30
Now plug b into the perimeter formula:
P = (8/3)(30)

- Now perform the multiplication:
P = 240/3
P = 80

### Conclusion
The perimeter of A’Leila’s pen is 80 yards.

If you need any further explanation or if there's anything else you'd like to clarify, feel free to ask!

Explain why you used that expression to evaluate the perimeter of the
donkey’s pen.

Answer 1

The expression used to evaluate the perimeter of A’Leila’s pen is based on the relationship between the lengths of the sides of the rectangle formed by the pen. Here’s the breakdown of why each step was important:

1. Understanding the Relationship Between the Sides: The problem states that one side of the rectangle (a) is one-third of the other side (b). This relationship, expressed as \( a = \frac{1}{3} b \), is crucial as it allows us to express one dimension in terms of the other. This is particularly useful because we could only determine one variable from the condition provided.

2. Using the Perimeter Formula: The formula \( P = 2a + 2b \) is essential since it defines how the perimeter of a rectangle is calculated. It states that the perimeter is the sum of all sides, which for a rectangle can be simplified to twice the sum of its length and its width.

3. Substituting \( a \) into the Perimeter Formula: By substituting \( a \) with \( \frac{1}{3} b \) in the perimeter formula, we can express the entire perimeter in terms of a single variable, \( b \). This simplifies our calculations, especially since we eventually have a specific value for \( a \) later on.

4. Simplifying the Expression: By performing the arithmetic and combining like terms, we arrive at \( P = \frac{8}{3}b \). This simplified expression allows us to compute the perimeter directly once we know the value of \( b \).

5. Finding \( b \) When \( a = 10 \): The step of substituting \( a \) with 10 and finding \( b \) is key. This step utilizes the initial relationship between the sides and allows us to find the corresponding length of \( b \) that works with our given \( a \).

6. Calculating the Perimeter: Finally, substituting \( b = 30 \) into the previously derived perimeter formula \( P = \frac{8}{3}b \) gives us the total perimeter of the pen. This step provides the final answer, which is critical for understanding the dimensions of A’Leila’s pen.

In conclusion, each step built upon the previous ones, using the geometric relationships and properties of a rectangle to arrive at the final perimeter calculation. This structured approach ensures that we efficiently and accurately determine the dimensions and subsequently the perimeter of the donkey’s pen. If you have more questions or need further clarification on any details, I’m here to help!

Answer 2

answer the question: Explain why you used that expression to evaluate the perimeter of the
donkey’s pen.

Answer 3

The expression used to evaluate the perimeter of the donkey’s pen is derived from the fundamental relationship between the sides of the rectangle. I used it for the following reasons:

1. Relationships: The problem establishes a specific relationship between the lengths of the sides of the pen, where one side (a) is one-third of the other side (b) represented by the equation \( a = \frac{1}{3}b \). This relationship is essential because it allows us to express one variable in terms of the other, simplifying our calculations.

2. Perimeter Formula: The perimeter \( P \) of a rectangle can be expressed with the formula \( P = 2a + 2b \). This formula provides a straightforward way to find the perimeter by summing all the sides. By using this expression, I can take advantage of the rectangle's properties to easily compute the total distance around the pen.

3. Substitution for Simplification: By substituting \( a \) with \( \frac{1}{3}b \) in the perimeter formula, I transformed the perimeter expression into a function of just one variable, \( b \). This is beneficial because we only needed to determine one side's length to find the perimeter. It simplifies the calculation, avoiding the need to evaluate two different variables independently.

4. Final Calculation: With the perimeter expressed as \( P = \frac{8}{3}b \), it became straightforward to substitute the value of \( b \) once it was calculated (in this case, derived from knowing \( a = 10 \)). This approach leads directly to the perimeter calculation without confusion over multiple variables.

In summary, I used that expression to efficiently evaluate the perimeter of the donkey’s pen, leveraging the relationship between the sides, the standard perimeter formula, and simplification through substitution, ultimately making it easier to arrive at an accurate answer.

Answer 4

answer the question: Explain why you used that expression ( a + b + a + b = 2a + 2b ) to evaluate the perimeter of the
donkey’s pen.

Answer 5

I used the expression \( a + b + a + b = 2a + 2b \) to evaluate the perimeter of the donkey’s pen because it captures the fundamental way to calculate the total distance around a rectangle. Here’s why this expression was appropriate and effective:

1. Definition of Perimeter: The perimeter of a rectangle is defined as the sum of all its sides. For a rectangle with length \( a \) and width \( b \), there are two sides of length \( a \) and two sides of length \( b \). Therefore, adding these sides together (i.e., \( a + b + a + b \)) accurately represents the total perimeter.

2. Simplification: The expression simplifies to \( 2a + 2b \), which allows us to factor out common terms. This simplification is helpful because it reduces the number of calculations we need to perform. Instead of adding four separate terms, we can combine them into a more manageable expression.

3. Clarity and Structure: Using the \( 2a + 2b \) formulation provides a clear framework for understanding how the dimensions of the rectangle relate to its perimeter. It organizes the information in a way that is easy to grasp and follow, making it straightforward to substitute known values for \( a \) and \( b \) later in the calculation.

4. Direct Application: The structure of \( 2a + 2b \) is commonly used in geometry, making it familiar and easy to apply. When we know the lengths of the sides (or can express them in relation to each other, as we did), we can directly calculate the perimeter without creating unnecessary complexity.

By using the expression \( a + b + a + b = 2a + 2b \), I was able to clearly and efficiently compute the perimeter of the donkey’s pen, ensuring accuracy while also making the calculation process straightforward.

Answer 6

just make it a paragraph man

Answer 7

I used the expression \( a + b + a + b = 2a + 2b \) to evaluate the perimeter of the donkey’s pen because it accurately represents the total distance around a rectangle. This expression reflects the fundamental definition of perimeter, which is the sum of all sides—two sides of length \( a \) and two sides of length \( b \). Simplifying this expression to \( 2a + 2b \) not only makes the calculation easier but also provides a clear and organized structure for understanding the relationship between the dimensions of the rectangle and its perimeter. This familiar format allows for straightforward substitution of known values, enabling a quick and efficient calculation of the perimeter while ensuring accuracy throughout the process.