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Question 4

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=(13)b
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Write two different expressions giving the perimeter of the donkey’s pen.

7 answers

To find the perimeter of a rectangle, the formula is given by:

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

In this case, the length is \(b\) yards and the width is \(a\) yards, where \(a = \frac{1}{3}b\).

We can express the perimeter \(P\) in two different ways:

  1. Using \(a\) and \(b\):

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

Substituting \(a\) with \(\frac{1}{3}b\):

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

So, the first expression for the perimeter is:

\[ P = \frac{8}{3}b \]

  1. Using only \(b\) and substituting \(a\):

Another way to express the perimeter directly in terms of \(b\):

Since \(a = \frac{1}{3}b\), we can still write the perimeter using the initial formula:

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

Replacing \(a\):

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

Thus, both expressions you can use for the perimeter of A’Leila's pen are:

  1. \(P = \frac{8}{3}b\)
  2. \(P = 2(a + b)\) (where \(a = \frac{1}{3}b\))
Choose a value for a that is greater than 5 and less than 15. Evaluate the perimeter of the pen for that value, using one of the expressions that you wrote.

Let's choose a value for \(a\) that is greater than 5 and less than 15. A suitable choice for \(a\) could be 10.

Now, we know that \(a = \frac{1}{3}b\). To find \(b\), we can rearrange the equation:

\[ b = 3a \]

Substituting \(a = 10\):

\[ b = 3 \times 10 = 30 \]

Now, we can evaluate the perimeter of the pen using one of the expressions we derived earlier. Let's use the expression:

\[ P = \frac{8}{3}b \]

Substituting \(b = 30\):

\[ P = \frac{8}{3} \times 30 \]

Calculating this:

\[ P = \frac{240}{3} = 80 \]

Thus, the perimeter of the pen for \(a = 10\) is 80 yards.

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

I used the expression \(P = \frac{8}{3}b\) to evaluate the perimeter of the donkey's pen for the following reasons:

  1. Direct Calculation: This expression provides a straightforward way to calculate the perimeter directly in terms of \(b\), which we determined after selecting a specific value for \(a\). Since \(b\) was calculated as \(30\) when \(a = 10\), substituting this value into the expression allowed for a quick and easy calculation of the perimeter.

  2. Derived from General Formula: The expression came from the general formula for the perimeter of a rectangle, and it is already in simplified form. It incorporates the relationship between \(a\) and \(b\) (\(a = \frac{1}{3}b\)), so it takes into account the dimensions of the pen while also simplifying the calculation process.

  3. Simplicity and Clarity: Using \(P = \frac{8}{3}b\) minimizes the need for additional substitutions or calculations, making it clearer and less error-prone when computing the perimeter. It combines both measurements into a single expression, which is beneficial for quick evaluations.

  4. Alignment with the Problem: Since the task was to evaluate the perimeter based on a chosen \(a\) while linking it to the corresponding \(b\), using an expression that relates directly back to the given relationships simplifies the process and keeps the solution consistent with the derived mathematical framework.

Overall, this expression allowed for efficient calculation while adhering closely to the defined relationships between the dimensions of the pen.

Tell me the answer.

The perimeter of the donkey's pen, using \(a = 10\) and \(b = 30\), is 80 yards.