Question

the irrigation design for a new backyard is shown with six sprinkler heads and two walkways. which statement is true of this design? the area covered by each sprinkler head is approximately 113.10 ft². the area covered by all the sprinkler heads together is approximately 169.64 ft². every square foot of the yard will be watered with this design. the sprinklers would cover more area if they were in a straight line.

Explanation:

Step1: Calculate the radius of each sprinkler head

From the grid, the radius of each sprinkler head \(r = 6\) feet (since the distance between grid lines is \(2\) feet and radius spans \(3\) grid - line intervals: \(3\times2=6\) feet).

Step2: Calculate the area of a single sprinkler head

The formula for the area of a circle is \(A=\pi r^{2}\). Substitute \(r = 6\) into the formula: \(A=\pi\times6^{2}=36\pi\approx36\times 3.14 = 113.04\approx113.10\) \(ft^{2}\).

Step3: Calculate the area of all six sprinkler heads

The area of all six sprinkler heads is \(6A=6\times36\pi = 216\pi\approx216\times3.14=678.24\) \(ft^{2}\), so the second option (about \(169.64\) \(ft^{2}\) for all) is wrong.

Step4: Analyze coverage

There are walk - ways in the design, so not every square foot of the yard will be watered (eliminating the third option).

Step5: Analyze the effect of straight - line arrangement

For a given number of circles (sprinklers), a non - linear (in this case, a more spread - out in a grid - like pattern considering the yard's shape) arrangement may cover more area than a straight - line arrangement (because in a straight - line, there is more overlap at the ends compared to a more distributed pattern in the grid - like setting for the yard's shape), so the fourth option is wrong.

Answer:

The area covered by each sprinkler head is approximately \(113.10\) \(ft^{2}\)