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

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.

• The shed blocks a right angle, so the rope sweeps \(360^\circ - 90^\circ = 270^\circ\)
• \(A = \frac{270}{360}\times\pi\times 5^2 = 18.75\pi\)
• \(A = 58.9\) m²

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.

• \(\frac34\pi\times 17^2 = 216.75\pi\)
• Rope left along the 14 m wall: 17 − 14 = 3 m, giving \(\frac14\pi\times 3^2 = 2.25\pi\)
• Rope left along the 9 m wall: 17 − 9 = 8 m, giving \(\frac14\pi\times 8^2 = 16.0\pi\)
• \(A = 235\pi = 738\) m²

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.

• \(\frac12\pi\times 12^2 = 72\pi\)
• Left: 12 − 4 = 8 m, giving \(\frac14\pi\times 8^2 = 16\pi\)
• Right: 12 − (14 − 4) = 2 m, giving \(\frac14\pi\times 2^2 = 1\pi\)
• \(A = 89\pi = 280\) m²

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.

• \(\frac34\pi\times 10^2 = 75\pi\)
• The rope reaches the end of the 8 m wall with 10 − 8 = 2 m left. It wraps round: \(\frac14\pi\times 2^2 = 1\pi\)
• The rope is shorter than the 12 m wall, so there is no second wrap.
• \(A = 76\pi = 239\) m²