A farmer uses an irrigation sprinkler system that pivots about the center of a circular field. Determine the length of the sprinkler if an area of 2340 square feet is watered after a pivot of 20°.
P.S: The formula is A=1/2 theta r^2
Please help me. Thank you.
To find the length of the sprinkler, we need to first determine the radius of the circular field.
The formula given is A = (1/2) * θ * r^2, where A is the area of the sector, θ is the angle in radians, and r is the radius.
Given: A = 2340 square feet and θ = 20°.
Converting the angle from degrees to radians, we get: θ = 20° * π/180° = 0.3491 radians.
Now, let's rearrange the formula to solve for the radius:
A = (1/2) * θ * r^2
Divide both sides of the equation by (1/2) * θ:
2A / (θ) = r^2
Take the square root of both sides of the equation:
√(2A / θ) = r
Substituting the given values:
√(2 * 2340 sq.ft / 0.3491 radians) = r
Simplifying:
r ≈ √(13424.9) ≈ 115.9 feet
Now, to find the length of the sprinkler, we need to multiply the radius by the angle in radians:
Length = θ * r
Length = 0.3491 radians * 115.9 feet
Length ≈ 40.42 feet
Therefore, the length of the sprinkler is approximately 40.42 feet.
To determine the length of the sprinkler, we can utilize the formula for the area of a sector of a circle, which is given by A = (1/2)θr², where A is the area, θ is the central angle in radians, and r is the radius of the circle.
In this case, we are given the area A as 2340 square feet and the central angle θ as 20° (which needs to be converted to radians). However, we do not have the radius r, which we need to find.
Let's start by converting the central angle θ from degrees to radians. We know that π radians is equal to 180 degrees. Hence, we can set up a proportion to find the angle in radians:
θ degrees / 180 degrees = θ radians / π radians
Substituting the given value for θ = 20 degrees:
20 degrees / 180 degrees = θ radians / π radians
Simplifying this proportion, we get:
20 / 180 = θ / π
Dividing both sides by 20:
1 / 9 = θ / π
Cross-multiplying:
θ = π / 9
Now that we have the angle in radians (θ = π / 9), we can substitute this value along with the given area (A = 2340 square feet) into the formula A = (1/2)θr² to solve for r.
2340 = (1/2)(π / 9)r²
Multiply both sides by 2 to get rid of the fraction:
4680 = (π / 9)r²
To isolate r², divide both sides by π / 9:
4680 / (π / 9) = r²
Simplify the right side by multiplying by the reciprocal:
4680 * (9 / π) = r²
Dividing 4680 by π and then multiplying by 9:
42120 / π = r²
Taking the square root of both sides to solve for r:
r = √(42120 / π)
Hence, the length of the sprinkler is approximately equal to √(42120 / π).
12
Θ is in radians
20º = (2 π / 18) rad
2340 = ½ (2 π / 18) r²
18 * 2340 / π = r²