Question

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. Write two different expressions giving the perimeter of the donkey’s pen.

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.

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

Answer 1

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. \( P = 8a \)

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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 \( P = 8a \):

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

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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 \( a \).