Grazing goat problems I: sectors round a shed Problem set A

Two stone walls of a farmyard meet at a corner at an angle of \(\frac{\pi}{3}\). A goat is tied to the corner by a 6 m rope and grazes outside the angle between the walls. Both walls are longer than 6 m. Find the area of grass the goat can graze.

A goat is tied by a 5 m rope to the corner of a shed that measures 9 m by 7 m. Find the area of grass the goat can graze.

A goat is tied by a 17 m rope to a corner of a shed that measures 14 m by 9 m. Find the area of grass it can graze.

A shed measures 14 m by 10 m. A goat is tied by a 12 m rope to a hook on a 14 m wall, 4 m from the nearest corner. Find the area of grass the goat can graze.

A goat is tied by a 10 m rope to a corner of a shed that measures 12 m by 8 m. The rope is longer than the short wall and shorter than the long wall. Find the area of grass the goat can graze.

A shed is 12 m wide and 9 m deep. A goat is tied by an 11 m rope to a hook X on the front wall, x m from the left-hand corner, where \(2 \le x \le 10\).

1. (a) Find an expression for the grazed area A in terms of x.
2. (b) Find A when x = 3.

A goat is tied by an 11 m rope to a hook on the front wall of a shed that is 12 m wide and 9 m deep. When the hook is x m from the left-hand corner, \(2 \le x \le 10\), the grazed area is \(A = \frac{\pi}{2}(x^2 - 12x + 182)\) m². The goat grazes 230 m². Find the possible values of x.

Use \(A = \frac{\pi}{2}(x^2 - 12x + 182)\) for \(2 \le x \le 10\). Find where the hook should be to give the goat (a) the most grass and (b) the least grass, and the area in each case.