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

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.

• The shed blocks a right angle, so the rope sweeps \(360^\circ - 90^\circ = 270^\circ\)
• \(A = \frac{270}{360}\times\pi\times 6^2 = 27\pi\)
• \(A = 84.8\) m²

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.

• \(\frac34\pi\times 13^2 = 126.75\pi\)
• Rope left along the 11 m wall: 13 − 11 = 2 m, giving \(\frac14\pi\times 2^2 = 1.0\pi\)
• Rope left along the 7 m wall: 13 − 7 = 6 m, giving \(\frac14\pi\times 6^2 = 9.0\pi\)
• \(A = 136.75\pi = 430\) m²

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.

• \(\frac12\pi\times 11^2 = 60.5\pi\)
• Left: 11 − 5 = 6 m, giving \(\frac14\pi\times 6^2 = 9\pi\)
• Right: 11 − (15 − 5) = 1 m, giving \(\frac14\pi\times 1^2 = 0.25\pi\)
• \(A = 69.75\pi = 219\) m²

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.

• \(\frac34\pi\times 12^2 = 108\pi\)
• The rope reaches the end of the 9 m wall with 12 − 9 = 3 m left. It wraps round: \(\frac14\pi\times 3^2 = 2.25\pi\)
• The rope is shorter than the 15 m wall, so there is no second wrap.
• \(A = 110.25\pi = 346\) m²