Answer:
Step-by-step explanation:
First, let's look at the pie-shaped segment defined by the 123° angle and the 18.6 m radius.
The area of the entire circle is given by A = πr² = 3.14159(18.6 m)², or A = 1086.865 m².
The area of the "pie slice" is (123/360) of that, or 371.346 m².
Next, we must find the area of the triangle and subtract this area from this 371.346 m²:
Use the Law of Cosines to find the area of this triangle. The Law of Cosines in appropriate form is
c² = a² + b² -2ab*cos C.
In this case C is 123° and a and b are both 18.6 m in length. Thus,
c² = (18.6 m)² + (18.6 m)² - 2(18.6 m)(18.6 m)*cos 123°
Thus,
c² = 691.92 m² - 691.92*cos 123°, or
= (691.92 m²)(1 - cos 123°)
= (691.92 m²)(1 - [-0.5446] ) = (691.92 m²)(1.5446) = 1068.767
Taking the square root of this result, we get c = +32.6920 m.
Subdivide the triangle shown to form a new triangle with hypotenuse 18.6 m, adjacent side 32.6920 m / 2, or 16.3460 and opposite side h.
Then 18.6² = h² + 16.3460², which leads to h²2 = 18.6² - 16.346², or 78.768.
Then h = √78.768 = 8.875 (m).
We now use this info to find the area of the illustrated triangle, using the formula A = (1/2)(base)(height). That comes out to:
A = (1/2)(16.3460 m)(8.875 m) = 72.535 m²
Finally, we subtract twice that from the area of the 'pie' to get the area of the yellow (shaded) area:
pie area - area of triangle shown in the diagram =
371.346 m² - 145.071 m² = 226.3 m² (to the nearest tenth)
