To determine the area that the sprinkler can spray, we first note that the sprinkler can spray water in a semicircular area due to its 180° rotation.
The formula for the area \( A \) of a circle is:
\[ A = \pi r^2 \]
where \( r \) is the radius. In this case, the radius is 15 feet (the distance the sprinkler can spray). Since the sprinkler covers only half of the circle (180°), we will use half of the circle's area for our calculation.
First, we calculate the area of the full circle:
\[ A_{\text{circle}} = \pi (15)^2 = \pi \times 225 \]
Now, let's find the area for the semicircle:
\[ A_{\text{semicircle}} = \frac{1}{2} \pi (15)^2 = \frac{1}{2} \times \pi \times 225 = \frac{225\pi}{2} \]
Now we can use the approximate value of \( \pi \approx 3.14159 \) to find the numerical area:
\[ A_{\text{semicircle}} \approx \frac{225 \times 3.14159}{2} \approx \frac{706.85775}{2} \approx 353.428875 , \text{ft}^2 \]
Therefore, the area sprayed by the sprinkler is approximately 353.43 ft². Since no answer choice matches this value, let's go through the available choices again.
However, it's important to note that none of the provided answer choices seem to match.
If we simplify more and assess only the provided options, we would need to check for any potential miscalculations or misinterpretations in the context as well. But under normal circumstances, the calculated area seems correct based on the described conditions.
So the answer to the area of the backyard that the sprinkler can spray is approximately 353.43 ft², but given the options, it seems none are correct.
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