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

Two walls of a barn yard meet at a corner at an angle of 1.2 radians. A cow is tied to the corner by a 7.5 m rope and grazes outside the angle between the walls. Both walls are longer than 7.5 m. Find the area of grass the cow can graze.

A llama is tied by a 6 m rope to the corner of a greenhouse that measures 10 m by 8 m. Find the area of grass the llama can graze.

A donkey is tied by a 13 m rope to a corner of a barn that measures 11 m by 7 m. Find the area of grass it can graze.

A summer house measures 15 m by 8 m. A ram is tied by an 11 m rope to a hook on a 15 m wall, 5 m from the nearest corner. Find the area of grass the ram can graze.

A pony is tied by a 12 m rope to a corner of a stable that measures 15 m by 9 m. The rope is longer than the short wall and shorter than the long wall. Find the area of grass the pony can graze.

A barn is 14 m wide and 10 m deep. A donkey is tied by a 12 m rope to a hook X on the front wall, x m from the left-hand corner, where \(2 \le x \le 12\).

1. (a) Find an expression for the grazed area A in terms of x.
2. (b) Find A when x = 5.

A donkey is tied by a 12 m rope to a hook on the front wall of a barn that is 14 m wide and 10 m deep. When the hook is x m from the left-hand corner, \(2 \le x \le 12\), the grazed area is \(A = \frac{\pi}{2}(x^2 - 14x + 218)\) m². The donkey grazes 280 m². Find the possible values of x.

Use \(A = \frac{\pi}{2}(x^2 - 14x + 218)\) for \(2 \le x \le 12\). Find where the hook should be to give the donkey (a) the most grass and (b) the least grass, and the area in each case.