A sprinkler on a golf green is set to spray water over a distance of 20 meters and to rotate through an angle of 160°. Find the area of the region that can be irrigated with the sprinkler. (Round your answer to two decimal places.)
I really have no idea how to go about this problem, so I'd appreciate any help anyone could give. Thanks in advance!
To find the area that can be irrigated with the sprinkler, we need to determine the shape of the region covered by the water spray. Since the sprinkler sprays water in a circular pattern, the region will be a sector of a circle.
First, let's find the area of a full circle with a radius of 20 meters. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
A = π(20)^2
A = π(400)
A ≈ 1256.64 square meters
Next, we need to determine the fraction of the full circle that the sprinkler covers. The sprinkler rotates through an angle of 160°, which corresponds to 160/360 = 4/9 of a full circle.
Finally, we can find the area of the region covered by the sprinkler by multiplying the area of the full circle by the fraction of the circle covered by the sprinkler.
Area of region covered by sprinkler = (4/9) * 1256.64
Area of region covered by sprinkler ≈ 558.40 square meters
Therefore, the area of the region that can be irrigated with the sprinkler is approximately 558.40 square meters.