Question

Please help me on this question

Answer 1

The area of the pasture accessible to the goat is the area of the circle of radius 3 m, less the area of the circular segment beyond the fence. Calculating the segment's area requires we know something about the angle subtended by the fence where it crosses the circle.

From trigonometry, you know that the cosine of an angle is the ratio of the adjacent side to the hypotenuse. That is, the cosine of half the subtended angle is 2/3, so the arc crossed by the fence has measure ...

... 2·arccos(2/3) ≈ 96.379° ≈ 1.6821 radians

The area of the segment is given by

... A = (1/2)r²·(α - sin(α)) . . . . . where α is the angle just found

The area beyond the fence is then ...

... A = (1/2)(3 m)²(1.6821 - sin(1.6821)) ≈ 3.0975 m²

The total area of a circle of radius 3 m is

... A = πr² = π(3 m)² ≈ 28.274 m²

Thus, the area accessible to the goat is

... 28.274 m² - 3.0975 m² ≈ 25.18 m²