Answer :
To draw an arc that is 60 feet long with a maximum crane reach of 21 feet, the construction worker needs to turn the crane by an angle of approximately 161.62 degrees. The area covered by the sprinkler that can shoot water up to 17 meters and rotate by 42 degrees is approximately 314.16 square meters.
To find the angle needed to draw the arc, we can use the concept of the arc length formula. The formula for finding the length of an arc is given by:
Arc length = r * θ
where r is the radius (maximum distance of the crane reach) and θ is the angle turned by the crane. Rearranging the formula, we can solve for the angle:
θ = Arc length / r
Plugging in the values, we have:
θ = 60 feet / 21 feet = approximately 2.857 radians
To convert radians to degrees, we can use the conversion factor: 180 degrees = π radians. Therefore:
θ = 2.857 * (180 / π) ≈ 161.62 degrees
So, the construction worker needs to turn the crane by an angle of approximately 161.62 degrees to draw the desired arc.
To find the area covered by the sprinkler, we can use the formula for the area of a sector. The formula is given by:
Area = (θ / 360) * π * [tex]r^2[/tex]
where θ is the angle turned by the sprinkler and r is the maximum distance it can shoot water.
Plugging in the values, we have:
Area = (42 / 360) * π * [tex](17 meters)^2[/tex]
Area ≈ 0.116 * 3.1416 * 289 square meters
Area ≈ 314.16 square meters
Therefore, the area covered by the sprinkler is approximately 314.16 square meters.
Learn more about arc length formula
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