<h3>
Answer:</h3>
818.4 in²
<h3>
Step-by-step explanation:</h3>
The area of a circular sector is given by ...
... A = (1/2)r²·θ . . . . . θ in radians
The area of the isosceles triangle with apex angle 150° is given by ...
... A = (1/2)r²·sin(θ)
Then the area of the shaded segment is ...
... A = (1/2)r²·(θ - sin(θ))
... A = (1/2)·(27.8 in)²(5π/6 -sin(5π/6)) ≈ 818.4 in²
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If you (erroneously) use 3.14 for π, you get 817.9 in². Some answer keys expect you to use that value, even though it does not have sufficient accuracy for this problem.
