a sprinkler system on a baseball diamond rotates 120 degrees and sprays water up to 35 meters. What is the area of the diamond that can be watered with the sprinkler
a = 1/2 r^2 θ
= 1/2 (35^2) (2pi/3)
= 1282.8
To find the area that can be watered with the sprinkler system on a baseball diamond, we need to determine the shape that is covered by the rotating sprinkler.
Assuming the sprinkler rotates in a circular pattern, we can consider the area covered as a sector of a circle. The angle of rotation, 120 degrees, indicates that the sprinkler covers one-third of the entire circle (360 degrees).
To calculate the area of the sector, we need to know the radius of the circular area covered by the sprinkler. Unfortunately, the given information does not provide the radius directly. However, we can use the given distance of water spray, 35 meters, to calculate the radius of the circular area.
In a circle, the circumference is calculated using the formula C = 2πr, where r is the radius. Since the sprinkler sprays water up to 35 meters, the circumference of this circle is 35 meters. Therefore, we have the equation:
35 = 2πr
Solving for r:
r = 35 / (2π)
r ≈ 5.58 meters
Now that we have the radius, we can find the area of the sector using the formula:
A = (θ/360) * π * r^2
where θ is the angle of rotation in degrees, π is a mathematical constant (approximately 3.14159), and r is the radius.
Substituting the values:
A = (120/360) * π * (5.58)^2
A ≈ (1/3) * 3.14159 * 31.1364
A ≈ 32.9842 square meters
Therefore, the area of the diamond that can be watered by the sprinkler system is approximately 32.9842 square meters.