To determine the area of the largest circular portion of the yard that Sergio can enclose with 72 feet of fencing, we start by recognizing that the circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. Sergio has 72 feet of fencing to use as the circumference, so we set up the equation:
\[ 2\pi r = 72 \]
Now we can solve for \( r \):
\[ r = \frac{72}{2\pi} = \frac{36}{\pi} \]
Next, we can find the area \( A \) of the circular play space using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi \left(\frac{36}{\pi}\right)^2 \] \[ A = \pi \left(\frac{1296}{\pi^2}\right) \] \[ A = \frac{1296}{\pi} \]
Now, calculating this numerically, using \(\pi \approx 3.14\):
\[ A \approx \frac{1296}{3.14} \approx 412.68 \text{ square feet} \]
Thus, the area of the largest portion of the yard that Sergio can enclose is approximately 412.68 square feet.
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