Question

The irrigation design for a new backyard is shown with four sprinkler heads and two walkways.

The distance between grid lines is 3 feet.
4 sprinklers are on 2 separate pieces of land. Each sprinkler waters a circular area with a diameter of 9 feet. The sprinklers are at least 6 feet from each other. The sprinklers are more than 3 feet away from the walkway.

Which statements about the design are true? Select three options.

Each sprinkler head is placed more than 3 feet from the walkway.
If two additional sprinkler heads are placed on the lawn, every square foot of the yard would be watered.
The total area watered by each sprinkler is approximately 63.6 square feet.
The total area watered by all the sprinklers is approximately 254 square feet.
If the sprinklers were placed in two straight lines on the yard, they would water a greater amount of the lawn.

Answer 1

Select:
- Each sprinkler head is placed more than 3 feet from the walkway. (True)
- The total area watered by each sprinkler is approximately 63.6 square feet. (True)
- If the sprinklers were placed in two straight lines on the yard, they would water a greater amount of the lawn. (True)

Reasons (brief):
- The problem text states the sprinklers are more than 3 ft from the walkway, so the first statement is true.
- Area of one sprinkler = πr^2 = π(4.5)^2 ≈ 63.62 ft^2, so ≈63.6 ft^2 is correct.
- The four circles as currently placed overlap (centers are at least 6 ft apart but less than the 9 ft needed to avoid overlap), so the union area is less than 4×63.6. Placing heads in two straight lines can increase distances between centers and reduce overlap, increasing the total watered area.

The other two statements are false:
- Adding two heads does not guarantee every square foot would be watered.
- “Total area watered by all the sprinklers ≈ 254 ft^2” is simply 4×63.6 and ignores overlap, so the actual watered area is less than 254 ft^2.