Answer:
Area_shaded_region = 2 * (√(1 - cos^2(4π/9))) - (16/9)π m^2
Explanation:
To find the area of the shaded region, we need to subtract the area of the sector from the area of the triangle formed by the radius and the two radii connecting to the endpoints of the central angle.
First, let's find the area of the sector:
The formula for the area of a sector is (θ/360) * π * r^2, where θ is the central angle and r is the radius.
Given that the radius is 2 m and the central angle is 160°, we have:
θ = 160°
r = 2 m
Converting the angle to radians:
θ_radians = (160° * π) / 180° = (8π/9) radians
Now, we can calculate the area of the sector:
Area_sector = (θ_radians / (2π)) * π * r^2
= (8π/9) / (2π) * π * 2^2
= (4/9) * π * 4
= (16/9)π m^2
Next, let's find the area of the triangle:
The formula for the area of a triangle is (1/2) * base * height.
The base of the triangle is equal to the length of the radius, which is 2 m.
The height of the triangle can be found using the formula h = r * sin(θ/2).
θ = 160°
r = 2 m
Converting the angle to radians:
θ_radians = (160° * π) / 180° = (8π/9) radians
Calculating the height:
h = 2 * sin(θ_radians/2)
= 2 * sin((8π/9)/2)
= 2 * sin(4π/9)
= 2 * (√(1 - cos(4π/9)^2))
= 2 * (√(1 - cos^2(4π/9)))
Now, we can calculate the area of the triangle:
Area_triangle = (1/2) * base * height
= (1/2) * 2 * 2 * (√(1 - cos^2(4π/9)))
= 2 * (√(1 - cos^2(4π/9)))
Finally, we can find the area of the shaded region by subtracting the area of the sector from the area of the triangle:
Area_shaded_region = Area_triangle - Area_sector
= 2 * (√(1 - cos^2(4π/9))) - (16/9)π m^2
This is the exact answer in terms of π, with the correct unit of measurement (m^2).