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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.

5 answers

To find the dimensions of the pen and its area based on the relationship \( a = \frac{1}{3}b \), we can express everything in terms of one variable.

  1. Dimensions of the Pen:

    • Let \( b \) be the length of one side of the pen (measured in yards).
    • Then, by the relationship given, the other side \( a \) can be expressed as: \[ a = \frac{1}{3}b \]

  2. Area of the Pen:

    • The area \( A \) of a rectangle is given by the formula: \[ A = a \times b \]
    • Substituting \( a = \frac{1}{3}b \) into the area formula: \[ A = \left(\frac{1}{3}b\right) \times b = \frac{1}{3}b^2 \]

Now, we have expressed both the dimensions and the area of A’Leila's pen in terms of \( b \).

If you have a specific value for \( b \) or would like to calculate a particular aspect of the pen (like the area or perimeter), please provide that value!

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. keep answer short

Let's choose \( a = 10 \) (which is greater than 5 and less than 15).

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

\[ 10 = \frac{1}{3}b \implies b = 30 \]

Now, we can calculate the perimeter \( P \) of the rectangle:

\[ P = 2(a + b) = 2\left(10 + 30\right) = 2 \times 40 = 80 \text{ yards} \]

Thus, the perimeter of the pen is 80 yards.

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

I used the expression \( P = 2(a + b) \) to evaluate the perimeter of the donkey's pen because it is the standard formula for calculating the perimeter of a rectangle. This formula adds the lengths of both adjacent sides and then doubles the sum to account for all four sides. Since we expressed one side in terms of the other, it allowed us to use a specific value for \( a \) and find the corresponding \( b \) easily.